Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Integrating a Rational Function with a Square Root : $ \int \frac{dx}{\sqrt{x^2-3x-10}} $
Integrate $$ \int \frac{dx}{\sqrt{x^2-3x-10}}. $$
I started off with the substitution $\sqrt{x^2-3x-10} = (x-5)t$. To which I got
$$ x = \frac{2+5t^2}{t^2-1} \implies \sqrt{x^2-3x-10} = \left(\frac{2+5t^2}{t^2-1} - 5\right)t = \... | See
$$- \ln\left| \frac{\sqrt{x^2-3x-10} - x + 5}{\sqrt{x^2-3x-10} + x - 5} \right| = \ln\left| \frac{\sqrt{x^2-3x-10} + x - 5}{\sqrt{x^2-3x-10} - x + 5} \right| $$
and
\begin{align*}
\frac{\sqrt{x^2-3x-10} + x - 5}{\sqrt{x^2-3x-10} - x + 5}
=&
\frac{\sqrt{x^2-3x-10} + (x - 5)}{\sqrt{x^2-3x-10} - (x - 5)}
\times
\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
The limit of a fraction What is the value of
$$\lim_{x\to\infty}\frac{\sqrt{x^4-1}}{2x^2+3x-1}$$
I tried to factorise the denominator or the numerator but it takes me nowhere. My teacher said it equals to $\frac{1}{2}$. But I don't get it. Can anyone help me?
| $$\lim_{x\rightarrow+\infty}\frac{\sqrt{x^4-1}}{2x^2+3x-1}=\lim_{x\rightarrow+\infty}\frac{\sqrt{1-\frac{1}{x^4}}}{2+\frac{3}{x}-\frac{1}{x^2}}=\frac{1}{2}$$
Also
$$\lim_{x\rightarrow-\infty}\frac{\sqrt{x^4-1}}{2x^2+3x-1}=\lim_{x\rightarrow-\infty}\frac{\sqrt{1-\frac{1}{x^4}}}{2+\frac{3}{x}-\frac{1}{x^2}}=\frac{1}{2}$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2434197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Why can complex numbers be written in exponential form? $z=r(\cos \theta+i\sin \theta)$ is $z=re^{i\theta}$. Why can complex numbers be written in exponential form? $z=r(\cos \theta+i\sin \theta)$ is $z=re^{i\theta}$.
I have studied that the exponential form of a complex number $z=r(\cos \theta+i\sin \theta)$ is $... | Lets consider a function from $\mathbb R\to \mathbb C$
$z(\theta) = \cos \theta + i\sin \theta\\
z(\theta)z(\phi) = (\cos \theta + i\sin \theta)(\cos \phi + i\sin \phi) = \cos(\theta + \phi) + i\sin (\theta+\phi) = z(\theta + \phi)$
That is a property of an exponential function. We do not know the base.
For some base:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2436350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 3
} |
Are the vectors linearly dependent or independent? Problem: Let $\vec{v}=(5,9), \ \vec{u}=(3,-2)$ and $\vec{w}=(2,1)$. Determine the nature of their linear dependency.
Attempt: So, we are looking for constants $k_1,k_2,k_3$ such that $k_1\vec{v}+k_2\vec{u}+k_3\vec{w}=0.$ We can write this as
$$k_1\left[\begin{matrix} ... | You have made a mistake, it should be:
$$M\vec{k}=\left[ {\begin{array}{cc}
5 & 3 & 2 \\
9 & -2 & 1 \\
\end{array} } \right]\cdot \left[\begin{matrix} k_1 \\ k_2 \\ k_3 \end{matrix}\right]=\left[\begin{matrix} 0 \\ 0 \end{matrix}\right]$$
The solutions are of the form $\alpha\cdot \left[\begin{matrix} -7 \\ -13 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2437485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
For each $a \in \Bbb{Z}$ work out $\gcd(3^{16} \cdot 2a + 10, 3^{17} \cdot a + 66)$ The problem is the following: For each $a \in \Bbb{Z}$ work out $\gcd(3^{16} \cdot 2a + 10, 3^{17} \cdot a + 66)$
This is what I have at the moment:
Let's call $d = \gcd(3^{16} \cdot 2a + 10, 3^{17} \cdot a + 66)$
Then $d \ \vert \ 3^{1... | The prime factorization of $102$ is $2 \times 3 \times 17$. When do $2$, $3$ and $17$ divide your numbers?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2437747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Proof that $lim_{x->0,y->0}{\frac {\sqrt {a+x^2y^2} -1} {x^2+y^2}} (a>0)$ doesn't exist while a $\ne 1$. How to prove that $\lim_{x\to 0,y\to 0}{\frac {\sqrt {a+x^2y^2} -1} {x^2+y^2}} (a>0)$ doesn't exist while a $\ne 1$?
I already calculated that when a = 1 by multiplying $\sqrt {a+x^2y^2} + 1$ on both denominator and... | Evaluating on the path $x=y$:
$$\lim_{x,y\to0}{\frac{\sqrt{a+x^2y^2}-1}{x^2+y^2}}=\lim_{x\to0}{\frac{\sqrt{a+x^4}-1}{2x^2}}=\lim_{x\to0}{\frac{\sqrt a-1}{2x^2}}$$
If $a\ne1$ then $\sqrt a-1\ne0$ and a singularity exists at $x=y=0$, so the limit does not exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2439744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Three dice are rolled simultaneously. In how many different ways can the sum of the numbers appearing on the top faces of the dice be $9$?
Three dice are rolled simultaneously. In how many different ways can the sum of the numbers appearing on the top faces of the dice be $9$?
What I did:
I know that the maximum valu... | We must find the number of solutions of the equation
$$x_1 + x_2 + x_3 = 9 \tag{1}$$
in the positive integers subject to the restrictions that $x_1, x_2, x_3 \leq 6$.
A particular solution of equation 1 in the positive integers corresponds to the placement of $3 - 1 = 2$ addition signs in the $9 - 1 = 8$ spaces between... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How to approach the divergence of $\sum\frac{n!}{(x-1)(x-2)...(x-n)}$ If $x>0$, prove that the following series is divergent:
$\sum\frac{n!}{(x-1)(x-2)\ldots(x-n)}$
where $n=1,2,3\ldots$
I have proved that the absolute values series is divergent, but I cannot establish an inequality between both series. Help me, please... | In another way
$$
\eqalign{
& t_n = {{n!} \over {\left( {x - 1} \right)\left( {x - 2} \right) \cdots \left( {x - n} \right)}} = {{n!} \over {\left( {x - 1} \right)^{\,\underline {\,n\,} } }} \cr
& {{t_{n + 1} } \over {t_n }} = {{n + 1} \over {\left( {x - n - 1} \right)}} = {x \over {x - n - 1}} - 1 \cr
& \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2442834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Find limit of trigonometric function $$\lim_{x\rightarrow0} \frac{\tan^3(3x)-\sin^3(3x)}{x^5}$$
I think it should be decomposed with $$\lim_{x\to0} \frac{\sin x}{x}=1$$ but I'm always getting indefinity $\frac{0}{0}$.
| $$\begin{align}&\lim_{x \to 0} \dfrac{\tan^3 3x - \sin^3 3x}{x^5} &\\= &\lim_{x \to 0} (\sin^3 3x)\dfrac{1 - \cos^3 3x}{x^5}&\\=\, &27 \lim_{x \to 0} \dfrac{(1 - \cos 3x)}{x^2}(1 + \cos 3x + \cos^2 3x) &\\=\, &81 \lim_{x\to 0} \dfrac{(1 - \cos 3x)}{x^2}\end{align}$$
Let $x = 2y$
$$\lim_{y \to 0} \dfrac{(1 - \cos 6y)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2444868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Let $G$ be a group and $a,b \in G$ show that $|b| = 5$
Show that if $|a|=2$, $ab^2a^{-1}=b^3$ and $b\ne e$, then $|b|=5$.
My attempt
$$a^2 = e \implies a * a = e \implies a = a^{-1}$$
$$ab^2a=b^3$$
$$(ab^2a)^2=(b^3)^2$$
I) $$ab^2aab^2a = b^5$$
$$ab^2a=b^3$$
$$ab^2ab^2=b^3b^2$$
II) $$ab^2ab^2 = b^5$$
I and II$\implies... | One has
$ab^2 = b^3a$, then $(ab^2)^2 = b^3ab^3a = b^3a(ab^3a)ba = b^6aba$
$b^5 = b^3b^2 = ab^2a^{-1}b^2=ab^2ab^2=(ab^2)^2 = b^6aba$.
So, $baba = e$, then $b^{-1} = aba$, or $b^{-2} = ab^2a = b^3$, then $b^5=e$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Complex quintic equation Given the equation $x^5=i$, I need to show by both algebraic and trigonometrical approaches that
$$\cos18^{\circ}=\frac{\sqrt{5+2\sqrt5}}{\sqrt[5]{176+80\sqrt5}}$$
$$\sin18^{\circ}=\dfrac1{\sqrt[5]{176+80\sqrt5}}$$
Trying by trigonometric approach,
$x^5$ = i $\;\;\;\;$ -- eqn. (a)
=> x = $... | I think you're going at this backwards. Quickly, $\arg \mathrm{i} = \pi/2 = 90^\circ$, so fifth roots of $\mathrm{i}$ are points on the unit circle at small multiples of $18^\circ$. If the given values of cosine and sine are to be believed, then \begin{align*}
\mathrm{i} &= \left( \cos 18^\circ + \mathrm{i} \sin 18^\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Seeking the maximal parameter value s.t. two-variable inequality still holds Consider the expression
$$\frac 12\left(\frac{a^2}b+\frac{b^2}a\right)$$
in two variables $\,a,b\,$ residing in $\,\mathbb R^{>0}$.
The arithmetic mean $\,\frac{a+b}2\,$ is a lower bound for it
$\big[$ one has $\,a^2b\le(2a^3+b^3)/3\,$ by AM-G... | Here is a proof that the maximal $x$ must be $\le 9$.
We consider $a = 1 + u$, $b = 1$ and compute the Taylor development of
$$
f(u) = \frac 12 \left( (u+1)^2 + \frac{1}{u+1}\right) -
\left( \frac{(1+u)^x + 1}{2}\right)^\frac 1x
$$
at $u = 0$.
From the geometric series
we get
$$
\frac 12 \left( (u+1)^2 + \frac{1}{u+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2447859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
How to evaluate the sum : $\sum_{k=1}^{n} \frac{k}{k^4+1/4}$ I have been trying to figure out how to evaluate the following sum:
$$S_n=\sum_{k=1}^{n} \frac{k}{k^4+1/4}$$
In the problem, the value of $S_{10}$ was given as $\frac{220}{221}$.
I have tried partial decomposition, no where I go. Series only seems like it t... | By Sophie Germain's identity
$$ 4k^4+1 = (2k^2+2k+1)(2k^2-2k+1) \tag{A}$$
hence
$$ \frac{1}{2k^2-2k+1}-\frac{1}{2k^2+2k+1} = \frac{4k}{4k^4+1} = \frac{k}{k^4+1/4}\tag{B} $$
and we may notice that by setting $p(x)=2x^2-2x+1$ we have $p(x+1)=2x^2+2x+1$.
In particular
$$ \sum_{k=1}^{n}\frac{k}{k^4+1/4}=\sum_{k=1}^{n}\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2449134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
Finding the Smallest Convex Hull of an Adjacency Matrix Let's say I have a an adjacency matrix of a directed graph, with at most one 1 in each row (self-loops are allowed but are 1, not 2). My goal is to rearrange the matrix using only row and column swaps. If I treat the positive entries as points, I can shrink the si... | This isn't a proof. I noticed that we can treat each row as an integer since it has at most one 1 in a row. The column position of the one gives us the value, with the zero row being 0. We can then sort numbers based on how many duplicates there are for that number. So, if we have 1,2,4,4,4,5,5,6,6,6 we can sort it to,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2449373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to compute symmetrical determinant I'm learning of determinants and am trying to find a trick to compute this one
\begin{pmatrix}
2 & 1 & 1 & 1 & 1\\
1 & 3 & 1 & 1 & 1\\
1 & 1 & 4 & 1 & 1\\
1 & 1 & 1 & 5 & 1\\
1 & 1 & 1 & 1 & 6
\end{pmatrix}
I expanded it out and got $349$ but I feel there must b... | There is a general pattern for this kind of matrices. Define
$$
A_n:=\mathbf 1_n+\operatorname{diag}(1,2,\dots,n)=\pmatrix{2&1&1&\dots &1&1\\
1&3&1&\dots&1&1\\
\vdots&\ddots&\ddots& & \vdots& \vdots\\
1&1&1&\dots &n&1\\
1&1&1&\dots &1&n+1},
$$
where $\mathbf 1_n$ is the $n\times n$ matrix whose entries are one.
View t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2449842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Determining whether a vector is an element of the row space. I've been having trouble with determining whether a vector is an element of the row space. Specifically, for the matrix:
$$\begin{bmatrix}-2 & 6 & 7 & -1 \\ 7 & -3 & 0 & -3 \\ 8 & 0 & 6 & 7\end{bmatrix}$$
Whether the vector:
$$\begin{bmatrix} 2 \\ 1 \\ 3 \\ -... | If your reduced row echelon form (including the augmented part) reduces to the identity matrix, it is not consistent.
For example $$\left[\begin{array}{c|c} 1 & 0 \\ 0 & 1\end{array}\right]$$
implies that $x=0$ and $\color{blue}{0=1}$ which is not consistent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2449946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Polynomial $P(x)$ with $P(x)\in\mathbb Q$ iff $x\in \mathbb Q$
Find all the polynomials $P \in \mathbb{R}[X]$ such that: $P(x) \in \mathbb{Q}$ if and only if $x \in \mathbb{Q}$.
I think $P(x)=x+c$ or $P(x)=-x+c$ where $c$ is some rational constant. But I have no idea approach this.
| Partial Answer: We can show that $P(x)$ can't be a quadratic with rational coefficients. In fact, $P(x)$ can't be a quadratic at all.
Let $P(x)=ax^2+bx+c$, for $a,b,c\in\mathbb{Q}$. Then $x=\frac{-b+\sqrt{m^2+1}}{2a}$, where $m\in\mathbb{Z}$, will be an irrational $x$ with a rational $P(x)$.
Constant coefficient, an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2450237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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Proof that $\binom{n + 3}{4} = n + 3 \binom{n + 2}{4} - 3 \binom{n + 1}{4} + \binom{n}{4}$. I was trying to count the number of equilateral triangles with vertices in an regular triangular array of points with n rows. After putting the first few rows into OEIS, I saw that this was described by A000332: $\binom{n}{4} = ... | This is calculus of finite differences:
$$\binom{n+1}2-\binom{n}2=\binom{n}1=n,$$
$$\binom{n+1}3-\binom{n}3=\binom{n}2,$$
$$\binom{n+1}4-\binom{n}4=\binom{n}3,$$
etc. Then
$$n=\binom{n+1}2-\binom{n}2
=\left(\binom{n+2}3-\binom{n+1}3\right)
-\left(\binom{n+1}3-\binom{n}3\right)
=\binom{n+2}3-2\binom{n+1}3+\binom{n}3
=\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
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If we know range of a function , how can construct range of other function?
If we know $\frac{-1}{2}\leq \frac{x}{x^2+1}\leq \frac{1}{2}$ how can obtain the range of the function below?
$$y=\frac{4 x}{9x^2+25}$$
The problem is what's the range with respect to $\frac{-1}{2}\leq \frac{x}{x^2+1}\leq \frac{1}{2}$
| Write
$$\frac{4x}{9x^2+25}=\frac{4}{9x+\frac{25}{x}}$$ and use AM-GM in two cases:
1) $x>0$:
$$y\leq\frac{4}{2\sqrt{9x\cdot\frac{25}{x}}}=\frac{2}{15}.$$
2) $x<0$: $$y\geq-\frac{4}{2\sqrt{(-9x)\cdot\left(-\frac{25}{x}\right)}}=-\frac{2}{15}.$$
The equality occurs for $9x=\frac{25}{x}$ and since we have a continuous fun... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Proof that the square of an odd number is of the form $8m+1$ for some integer $m$
Show that $n$ is odd $\rightarrow \exists m \in \mathbb Z,n^2 = 8m + 1$
Let $k$ be an integer so that $n = 2k+1$. Then $n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2\left(2k^2+2k\right) + 1$.
But this isn't equal to $8m+1$, so can I change that ... | $$n^2 = (2k+1)^2 = 4k^2+4k+1 = 4k(k+1)+1$$
At least one of $k$ and $k+1$ is divisible by $2$, so let $k(k+1)=2m$. Then, $n^2 = 4(2m)+1 = 8m+1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Basis for Kernel and Image of a linear map I have to find the basis for the kernel and image of the following linear map.
$ \phi: R^3 → R^2, ϕ \begin{pmatrix} \begin{pmatrix} x \\y \\z \end{pmatrix}\end{pmatrix}= \begin{pmatrix} x -y \\z \end{pmatrix} $
For the range, I think we can express any arbitrary linear transfo... | Your calculations are correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find all n for which $2^n + 3^n$ is divisible by $7$? I am trying to do this with mods. I know that:
$2^{3k+1} \equiv 1 \pmod 7$, $2^{3k+2} \equiv 4 \pmod 7$, $2^{3k} \equiv 6 \pmod 7$ and $3^{3k+1} \equiv 3 \pmod 7$, $3^{3k+2} \equiv 2 \pmod 7$, $3^{3k} \equiv 6 \pmod 7$, so I thought that the answer would be when $... | Since $(x-2)(x-3)=x^2-5x+6$, we get that
$$
a_n=5a_{n-1}-6a_{n-2}
$$
is satisfied by $a_n=2^n+3^n$. Since $a_0\equiv2\pmod7$ and $a_1\equiv5\pmod7$, we get the following sequence mod $7$:
$$
\color{#C00}{2},\color{#C00}{5},6,\color{#090}{0},6,2,\color{#C00}{2},\color{#C00}{5},
$$
Thus, the sequence mod $7$ has period $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2454402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
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Then find the value of ... Let $a, \: b, \: c$ be three variables which can take any value (Real or Complex). Given that $ab + bc + ca = \frac{1}{2}$; $a + b + c = 2$; $abc = 4$. Then find the value of $$\frac{1}{ab + c - 1} + \frac{1}{bc + a - 1} + \frac{1}{ac + b - 1}$$
I always get stuck in such problems. Please giv... | Hint:
Let $y=\dfrac1{ab+c-1}$
$\implies y=\dfrac 1{\dfrac12--bc-ca+c-1}=\dfrac2{2c(1-a-b)-1}=\dfrac2{2c(c-1)-1}=\dfrac2{2c^2-2c-1} $
$$\iff2y c^2-2y c-(y+2)=0\ \ \ \ (1)$$
Again $y=\dfrac1{ab+c-1}=\dfrac c{4+c^2-c}$
$$\iff c^2y-c(y+1)+4y=0\ \ \ \ (2)$$
Solve $(1),(2)$ for $c,c^2$ and use $c^2=(c)^2$ to form a cubic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Binomial theorem relating proof There is this identity
$$1 -\frac{1}{2}\binom{n}{1}+\frac{1}{3} \binom{n}{2}- \frac{1}{4}\binom{n}{3}+....+(-1)^n \frac{1}{n+1}\binom{n}{n}$$
And we are supposed to prove it using these two identities
$$k\binom{n}{k} = n\binom{n-1}{k-1}$$
and
$$\binom{n}{0} + \binom{n}{1} + \binom{n}{... | \begin{eqnarray*}
\frac{1}{i+1}= \int_0^1 x^i dx
\end{eqnarray*}
Sub this into the sum & interchange the order of the sum & the integral
\begin{eqnarray*}
\sum_{i=0}^{n} (-1)^i\binom{n}{i} \frac{1}{i+1}&=& \int_0^1 \sum_{i=0}^{n} (-1)^i\binom{n}{i}x^i dx \\
&=& \int_0^1 (1-x)^n dx \\
&=& \left[ \frac{-(1-x)^n}{n+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Solve equation with a conjugate How to solve this equation:
$$
|z|^2 - 9 - \bar z + 3i = 0
$$
I know that:
$$
|z|^2=z \bar z
$$
$$
|z|=\sqrt{a^2+b^2}
$$
$$
z = a + bi \Rightarrow \bar z = a - bi
$$
But I still don't know to calculate z from that.
| Be $z=x+iy$ so $|z|^2=x^2+y^2$, $\bar z=x-iy$. Replacing in the equation:$$|z|^2-9-\bar z+3i=0$$ $$x^2+y^2-9-x+iy+3i=0$$
Reagrouping,$$(x^2+y^2-9-x)+i(y+3)=0$$
So $Re(z)=0$ and $Im(z)=0$,$$y+3=0$$$$x^2+y^2-9-x=0$$
Solving for $y$ gives $y=-3$. Replacing in the other equation:
$$x^2+(-3)^2-9-x=0$$
$$x^2-x=0$$
Gives solu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Recurrence relation for the number of codewords of length $n$ from {0, 1, 2} with no two $0$’s appearing consecutively I'm supposed to use recurrence relations to find the number of codewords $b_n$ of length $n$ from the alphabet $\{0, 1, 2\}$ such that no two $0$’s appear consecutively.
After reading about this online... | The simple answer is that given a codeword of length $n-1$ we can add $10$ or $20$ to the end. Given a code word of length $n$ we can add $1$ or $2$ to the end. All code words of length $n+1$ can be gotten this way uniquely. So: $$b_{n+1}=2b_n+2b_{n-1}$$
There is a nice nerdy general approach to this sort of problem. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solve $2\log_bx + 2\log_b(1-x) = 4$ I need to solve $$2\log_bx + 2\log_b(1-x) = 4.$$
I have found two ways to solve the problem. The first (and easiest) way is to divide through by $2$:
$$\log_bx + \log_b(1-x) = 2.$$ Then, combine the left side: $$\log_b[x(1-x)] = 2,$$ and convert to the equivalent exponential form, $$... | You have to see that your equation $b^2=x(1-x)$ it has two unknowns actually so there are more than one solution. In fact you have the equation of the circle
$$(x-\frac12)^2+y^2=(\frac12)^2$$
The conclusion is that all the points $(x,b)$ in this semi-circle with $b$ positive are solutions of your problem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2458109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Proofs Involving Real Numbers We have the following rules that must be used in solving the question
*
*$a>0$ and $b>c$ $\implies$ $ab>ac$
*$a<0$ and $b>c$ $\implies$ $ab<ac$
*$a>b$ and $b>c$ $\implies$ $a>c$
Prove that the following is true: $\forall x,y \in \mathbb{R}$ is true that
$$ x>y \implies x... | Another way:
$\begin{array}\\
x^3-y^3
&=(x-y)(x^2+xy+y^2)\\
&=\frac12(x-y)(2x^2+2xy+2y^2)\\
&=\frac12(x-y)(x^2+2xy+y^2+x^2+y^2)\\
&=\frac12(x-y)((x+y)^2+x^2+y^2)\\
\end{array}
$
so
$x^3-y^3$
has the same sign as
$x-y$.
I prefer this
rewriting of
$x^2+xy+y^2$
as
$\frac12((x+y)^2+x^2+y^2)$
(rather than
$(x+y/2)^2+3y^2/4$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2459372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Finding the value of the definite integral $\int_0^2{x\int_x^2{\frac{dy}{\sqrt{1+y^3}}}}dx$ If $$f(x) = \int_x^2{\frac{dy}{\sqrt{1+y^3}}}$$
then find the value of $$\int_0^2{xf(x)}dx$$
I have no idea how to solve this question. Please help.
| The integration domain can be equivalently written as
$$
\Omega = \{(x,y): x<y<2 ~~\mbox{and}~~~ 0 < x < 2 \}
$$
or
$$
\Omega = \{(x,y): 0<x<y ~~\mbox{and}~~~ 0 < y < 2 \}
$$
Such that
\begin{eqnarray}
\int_0^2{\rm d}x\int_{x}^2{\rm d}y ~\frac{x}{\sqrt{1 + y^3}} &=& \int_0^2{\rm d}y\int_{0}^y{\rm d}x ~\frac{x}{\sqrt{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2460221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Finding integral solutions of $x+y=x^2-xy+y^2$
Find integral solutions of $$x+y=x^2-xy+y^2$$
I simplified the equation down to
$$(x+y)^2 = x^3 + y^3$$
And hence found out solutions $(0,1), (1,0), (1,2), (2,1), (2,2)$ but I dont think my approach is correct . Is further simplification required? Is there any other met... | Write $\Delta\geq0$.
It must help!
$$x^2-(y+1)x+y^2-y=0,$$
which gives $$(y+1)^2-4(y^2-y)\geq0$$ or
$$3y^2-6y-1\leq0$$ or
$$1-\frac{2}{\sqrt3}\leq y\leq1+\frac{2}{\sqrt3},$$
which gives $$0\leq y\leq2,$$
which gives all solutions:
For $y=0$ we get $x^2-x=0$, which gives $(0,0)$ and $(1,0)$.
For $y=1$ we get $x^2-2x=0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2460474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Verification of epsilon-delta proof I'd like to write an epsilon-delta proof that $$\lim_{x\to0} \frac{1}{x^2-1} = -1$$
My strategy in doing the proof was to choose $\delta < 1$ and then bound the $|x+1|$ and $x^2$ in $$ \left|\frac{1}{x^2-1} + 1\right| = \frac{x^2}{|x+1||x-1|}$$ If my math is correct, this yields $\fr... | Let $\varepsilon>0$ and $x\in (-1,1)$
$$\left|\frac{1}{x^2-1} + 1\right| <\varepsilon\iff \left|\frac{x^2}{x^2-1} \right|<\varepsilon \iff \left|-\frac{x^2}{1-x^2} \right|<\varepsilon \iff \frac{x^2}{1-x^2}<\varepsilon $$
$\iff x^2<\varepsilon -x^2\varepsilon \iff x^2(1+\varepsilon)<\varepsilon\iff |x|<\sqrt{\dfrac{\va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2462635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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How do you simplify an expression involving fourth and higher order trigonometric functions? The problem is as follows:
Which value of $K$ has to be in order that $R$ becomes independent from $\alpha$?.
$$R=\sin^6\alpha +\cos^6\alpha +K(\sin^4\alpha +\cos^4\alpha )$$
So far I've only come up with the idea that the solu... | Recall that
$$\sin^2\alpha = 1 - \cos^2\alpha$$
and express everything in terms of $\cos^2\alpha$:
$$\begin{align}
R &= \left(\;\sin^2\alpha\;\right)^3 + \left(\;\cos^2\alpha\;\right)^3+K\left(\;\left(\;\sin^2\alpha\;\right)^2+\left(\;\cos^2\alpha\;\right)^2\;\right) \\
&= \left(\;1-\cos^2\alpha\;\right)^3 + \left(\;\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 6,
"answer_id": 1
} |
Simplification of an expression with products $\prod_{1\leq iThe determinant of a matrix $C=(c_{ij})_{n\times n}$ whose entries have the form $c_{ij}=\frac{1}{a_i+b_j}$ is given by
$$\det C=\frac{\prod_{1\leq i<j\leq n}(a_i-a_j)(b_i-b_j)}{\prod_{1\leq i,j\leq n}(a_1+b_i)}.$$
In these notes (p. 145), this formula is app... | We have
$$\frac{\det G_m}{\det G}=\frac{\prod^{\prime}_{1\leq i<j\leq n}(i^2\pi^2-j^2\pi^2)^2}{\prod_{1\leq i<j\leq n}(i^2\pi^2-j^2\pi^2)^2}
\cdot
\frac{\prod_{1\leq i,j\leq n}(i^2\pi^2+j^2\pi^2)}{\prod^{\prime}_{1\leq i,j\leq n}(i^2\pi^2+j^2\pi^2)}.$$
And
\begin{align*} \frac{\prod^{\prime}_{1\leq i<j\leq n}(i^2\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Quadrilateral in Square
$S$ is a unit square. Four points are taken randomly, one on each side of $S$. A quadrilateral is drawn. Let the sides of this quadrilateral be $a,b,c,d$. Prove that $2\leq{}a^2+b^2+c^2+d^2\leq{}4$.
My Efforts:
Let
$\begin{align}m^2+t^2&=a^2&\mathfrak{a}\\n^2+o^2&=b^2&\mathfrak{b}\\p^2+q^2&=... | Since by AM-GM
$$mn+op+qr+ts\leq\left(\frac{m+n}{2}\right)^2+\left(\frac{o+p}{2}\right)^2+\left(\frac{q+r}{2}\right)^2+\left(\frac{t+s}{2}\right)^2=1,$$ we obtain:
$$a^2+b^2+c^2+d^2=4-2(mn+op+qr+ts)\geq4\cdot1-2=2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2467024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Non linear recursive sequence bounds: $14\le R_{100} \le 18$. Let $R_{n+1}=R_n+\frac{1}{R_n}$, where $R_1=1$.
I need to prove that $14\le R_{100} \le 18$.
Can anyone help please?
| Since $$R_n^2=R_{n-1}^2+\frac{1}{R_{n-1}^2}+2,$$ we obtain
$$R_{100}^2=R_1^2+\frac{1}{R_{99}^2}+...+\frac{1}{R_1^2}+2\cdot99>199>14^2.$$
Since, $R_{10}^2>2\cdot9+1=19,$ we obtain
$$R_{100}^2=\frac{1}{R_{99}^2}+...+\frac{1}{R_1^2}+199<\frac{89}{R_{10}^2}+\frac{10}{R_1^2}+199=\frac{89}{19}+10+199<324.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2469426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to find $\lim_{x \to 1^-} (x+1) \lfloor \frac{1}{x+1}\rfloor $? find the limits :
$$\lim_{x \to 1^-} (x+1) \lfloor \frac{1}{x+1}\rfloor =?$$
My try :
$$\lfloor \frac{1}{x+1}\rfloor=\frac{1}{x+1}-p_x \ \ \ : 0\leq p_x <1$$
So we have :
$$\lim_{x \to 1^-} (x+1) (\frac{1}{x+1}-p_x) \\= \lim_{x \to 1^-} -(x+1)p_x=!??$... | For $x$ near $1$ let say $$\frac12<x<1 \Longleftrightarrow -\frac12<x-1<0$$ we have have that
$$ \frac12<x<1 \Longleftrightarrow \frac32<x+1<2 \Longleftrightarrow \frac12 <\frac{1}{x+1} <\frac 23$$
Therefore
$$ \left\lfloor \frac{1}{x+1}\right\rfloor =0~~~\forall~ \frac12<x<1 $$
That is ,
$$ (x+1)\left\lfloor \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Integrate $\arctan{\sqrt{\frac{1+x}{1-x}}}$ I use partial integration by letting $f(x)=1$ and $g(x)=\arctan{\sqrt{\frac{1+x}{1-x}}}.$ Using the formula:
$$\int f(x)g(x)dx=F(x)g(x)-\int F(x)g'(x)dx,$$
I get
$$\int1\cdot\arctan{\sqrt{\frac{1+x}{1-x}}}dx=x\arctan{\sqrt{\frac{1+x}{1-x}}}-\int\underbrace{x\left(\arctan{\sq... | Continuing from
$$D=x\cdot\frac{1}{1+\frac{1+x}{1-x}}\cdot\frac{1}{2\sqrt{\frac{1+x}{1-x}}}\cdot\frac{1}{(x-1)^2},$$
and observing that $(x-1)^2 = (1-x)^2$, you can take the two factors of $(1-x)$ from the final denominator and multiply them into the first two fractions. This gives
$$\begin{align}
D&=x\cdot\frac{1}{(1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
In a triangle, $\Delta ABC$ we are given that $3 \sin A + 4 \cos B = 6$ and $4 \sin B + 3\cos A = 1$. Then what is the measure of angle $C$?
In a triangle, $\Delta ABC$ we are given that $3 \sin A + 4 \cos B = 6$ and $4 \sin B + 3\cos A = 1$. Then what is the measure of angle $C$?
On squaring and adding both the equa... | Note that $\sin(A+B) = \frac{1}{2}$ has two solutions in $(0,\pi)$, that are $A+B \in \{\frac{\pi}{6},\frac{5\pi}{6}\}$.
Now check which of them satisfies original equations. Lets suppose $A+B = \frac{\pi}{6}$. In triangle, all angles are positive, so both $0 < A,B \lt \frac{\pi}{6}$. This implies that
*
*$0 < \si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2479862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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What is the sum of $E=\frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\frac{4}{3^{4}}+...$ What is the right way to assess this problem?
$$E=\frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\frac{4}{3^{4}}+...$$
To find the value of the sum I tried to use the fact that it could be something convergent like a geometric series. H... | You can split it up into a sum of geometric progressions:
\begin{align} 3^{-1} + 2 \cdot 3^{-2} + 3 \cdot 3^{-3} + \dotsb = 3^{-1} + 3^{-2} + 3^{-3} + \dotsb \\
+ 3^{-2} + 3^{-3} + \dotsb \\
+ 3^{-3} + \dotsb \\
\ddots \\
= 1(3^{-1}+ 3^{-2} + 3^{-3} + \dotsb) \\
+3^{-1}(3^{-1}+ 3^{-2} + \dotsb) \\
+ 3^{-2}(3^{-1} + \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2480080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Find $\alpha + \beta$ given that: $\alpha^3-6\alpha^2+13\alpha=1$ and $\beta^3-6\beta^2+13\beta=19$ , ( $\alpha , \beta \in \Bbb R)$
Find $\alpha + \beta$ given that: $\alpha^3-6\alpha^2+13\alpha=1$ and $\beta^3-6\beta^2+13\beta=19$ , ( $\alpha , \beta \in \Bbb R)$
I have a solution involving a variable change ($\alp... | Lets create a polynomial for both these equations.
Let $A(x) = x^3 - 6x^2 +13x -1 $. and $B(x) = A(x) -18$. Both these polynomials have only one real root, which is easily verified from their derivative.
Now let $\alpha+\beta = a$. Then $\beta = a-\alpha$. So transforming $A(x)$ such that $x \to a-x$ should give a pol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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How to find the maximum and minimum values of $\frac{8x(x^2-1)}{(x^2+1)^2}$ algebraically? The function is $f(x) = \frac{8x(x^2-1)}{(x^2+1)^2}$.
I have tried using calculus, only to fail.
| HINT: prove that $$-2\le \frac{8x(x^2-1)}{(x^2+1)^2}\le 2$$
we have $$2-\frac{8x(x^2-1)}{(x^2+1)^2}=2\,{\frac { \left( {x}^{2}-2\,x-1 \right) ^{2}}{ \left( {x}^{2}+1
\right) ^{2}}}
$$
and $$2+\frac{8x(x^2-1)}{(x^2+1)^2}=2\,{\frac { \left( {x}^{2}+2\,x-1 \right) ^{2}}{ \left( {x}^{2}+1
\right) ^{2}}}
$$
the Minimum wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Show that $a_n=\frac{n+1}{2^{n+1}}\left(\frac{2}{1}+\frac{2^2}{2}+\frac{2^3}{3}+..+\frac{2^n}{n}\right)$ converges and find its limit Sequence: $$a_n=\frac{n+1}{2^{n+1}}\left(\frac{2}{1}+\frac{2^2}{2}+\frac{2^3}{3}+..+\frac{2^n}{n}\right)$$
My Attempt: I showed the sequence is increasing by considering the difference $... | $$ \lim_{n\to +\infty}\frac{\sum_{k=1}^{n}\frac{2^k}{k}}{\frac{2^{n+1}}{n+1}}\stackrel{\text{Cesàro-Stoltz}}{=}\lim_{n\to +\infty}\frac{\frac{2^{n+1}}{n+1}}{\frac{2^{n+2}}{n+2}-\frac{2^{n+1}}{n+1}}=\color{red}{1}, $$
no major mystery here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Proving the double differential of z = -z implies z= sinx In common usage, we know that $$\frac{d^2z}{dx^2}=-z\,$$ implies $z$ is of the form $a\sin x + b\cos x$. Is there a proof for the same. I was trying to arrive at the desired function but couldn't understand how to get these trigonometric functions in the equatio... | We assume the solution is of the form
$$z=\sum_{n=0}^{\infty}a_nx^n=a_0+a_1x+a_2x^2+\cdots+a_nx^n+\cdots$$
then
$$
z''
=\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}
=1\cdot2a_2+2\cdot3a_3x+3\cdot4a_4x^2+\cdots
$$
from $z''=-z$ then with arbitrary $a_0$ and $a_1$ we have
\begin{eqnarray*}
&&
a_2=-\frac{1}{1\times2}a_0 ~~~,~~~... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2485497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
max and min of $f(x,y)=y^8-y^4x^6+x^4$ The function is continuous in $\mathbb{R}^2$ and $f(x,y)=f(x,-y)=f(-x,y)=f(-x,-y)$.
If I consider $f(x,0)=x^4$ for $x\rightarrow +\infty$ $f$ is not limited up so $\sup f(x,y)=+\infty$.
But the origin is absolute min?
$f(x,x)=x^8-x^{10}+x^4\rightarrow -\infty$ if $x\rightarrow \i... | $f'_x=-6x^5 y^4+4x^3;\;f'_y=8y^7-4y^3x^6$
They must be both zero
$
\left\{
\begin{array}{l}
-2 x^3 (3 x^2 y^4-2)=0 \\
-4 y^3 \left(x^6-2 y^4\right)=0 \\
\end{array}
\right.
$
$(0,0)$ is a solution and then
$
\left\{
\begin{array}{l}
3 x^2 y^4-2=0 \\
x^6-2 y^4=0 \\
\end{array}
\right.
$
Solve for $y$ the first equat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2490870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Determing Fourier series and what it converges to I am trying to determine the fourier series of $f(x)$ on $[-2,2]$ which I believe I have done correctly. I will show some details below of my calculation. However my question is how do I determine what the Fourier series of $f(x)$ on $[-2,2]$ converges to. My guess is t... | Since $f (x) $ is piecewise smooth. But has a jump discontinuity at $x=0$ , the Fourier series will converge to $$1/2[ f (0-)+f (0+)]$$.
You don't need to calculate the series to find wheter it converges or not.
You can read more about it at http://tutorial.math.lamar.edu/Classes/DE/ConvergenceFourierSeries.aspx
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Calculus inequality involving sine and cosine community! I saw the following inequality in a calculus assignment, which I thought was harder to prove than I expected: For every $x\in\mathbb{R},$$$(\sin x + a \cos x)(\sin x + b \cos x)\leq 1+(\frac{a+b}{2})^2.$$
By means of a graphing device, I noticed that when $a=b$, ... | $$|\sin x+a\cos x|=\left|\sqrt{1+a^2}\left( \frac{1}{\sqrt{1+a^2}}\sin x+\frac{a}{\sqrt{1+a^2}}\cos x \right)\right|=|\sqrt{1+a^2}(\cos \theta \sin x+\sin \theta \cos x)|=|\sqrt{1+a^2}( \sin(x+\theta) )|\le \sqrt{1+a^2}.$$ So the maximum value of $\sin x+a\cos x=\sqrt{1+a^2}.$
\begin{align*}
|(\sin x+a\cos x)(\sin x+b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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In an Acute angled Triangle $\Delta ABC$ $\csc \left(\frac{A}{2}\right)+\csc \left(\frac{B}{2}\right)+\csc \left(\frac{C}{2}\right)=6$ In an Acute angled Triangle $\Delta ABC$ if $$\csc \left(\frac{A}{2}\right)+\csc \left(\frac{B}{2}\right)+\csc \left(\frac{C}{2}\right)=6$$ Prove that the Triangle is Equilateral.
I tri... | Let $f(x)=\frac{1}{\sin\frac{x}{2}}$.
Thus, $f''(x)=\frac{3+\cos{x}}{8\sin^3\frac{x}{2}}>0$ for all $x\in\left(0,\frac{\pi}{2}\right)$ and by Jensen we obtain:
$$\sum_{cyc}\frac{1}{\sin\frac{\alpha}{2}}\geq\frac{3}{\sin\frac{\alpha+\beta+\gamma}{6}}=6.$$
The equality occurs for $\alpha=\beta=\gamma=60^{\circ}$ only an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Induction for $3^n\geq n^2$ I need to prove by induction that $3^n\geq n^2$
What I have so far:
*
*Basis: $n = 1 \rightarrow 3^1 \geq 1^2 \rightarrow 3 \geq 1$
*Condition: $3^n\geq n^2$
*Assumption: $3^{n+1}\geq (n+1)^2$
*Basis: $3^n\geq n^2$
$3^{n+1}\geq (n+1)^2
= 3(3^n) \geq n^2+2n+1 $
That's where I got stuck... | We verify that it holds for $n=2$ since $3^2\ge2^2$
We assume that it holds for $n:$ $3^n\ge n^2(*)$ and we want to show that $3^{n+1}\ge(n+1)^2$
From $(*)$ we have by multiplying with $3$:
$3^{n+1}\ge3n^2$
But
$3n^2\ge(n+1)^2$ since $3n^2-(n+1)^2=2n^2-2n-2=n^2-2+(n-1)^2\ge0$
Since $n^2-2\ge0, (n-1)^2\gt0$ $\forall n\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2492240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
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Showing that there are no $3$-digit primes $\overline {abc}$ such that $b^2-4ac=9$.
Problem: Show that there are no $3$-digit primes $\overline {abc}$ such that $b^2-4ac=9$.
Solution. Upon the given condition, the quadratic equation $ax^2+bx+c$ can be written as $(px+q)(rx+s)$ and we have:
$$\overline{abc}=100a+1... | $c$ can take only $1, 3, 7, 9$ . [$abc$ is prime]
$b$ can take only $1, 2, 3, 4, 5, 6, 7, 8, 9$. [$b\neq 0$ as $-4ac$ can't be a square of a number]
$a$ can take $1, 2, 3, 4, 5, 6, 7, 8, 9$.
$b^2-k^2=4ac$.
Thus $b$ and $k$ both should be both odd or both even and $b>k$ as $c$ can't be $<0$.
$4ac=\text{Multiple of 4,12,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2493861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Find the limit of the complex function. $$ \lim_{x\to a} \left(2- \frac{x}{a}\right)^{\left(\tan \frac{\pi x}{2a}\right)}.$$
I have simplified this limit to this extent :
$$e^{ \lim_{x\to a} \left(\left(1- \frac{x}{a}\right){\left(\tan \frac{\pi x}{2a}\right)}\right)}$$
I don't know how to simplify the limit after that... | Let $y=(2- \dfrac xa)^{\tan \frac{\pi x}{2a}}$ then $\ln y=\tan\dfrac{\pi x}{2a}\ln(2- \dfrac xa)$ and let $1-\dfrac{x}{a}=\dfrac{1}{t}$ then
\begin{align}
\lim_{x\to a} (2- \dfrac xa)^{\tan \frac{\pi x}{2a}}
&= \lim_{t\to \infty} \tan\dfrac{\pi}{2}\left(1-\dfrac1t\right)\ln\left(1+\dfrac1t\right) \\
&= \lim_{t\to \inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 4
} |
Showing the range of a function The function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ is defined by
$$f(n)=\begin{cases}3n+2\quad \text{if $n$ is even}\\ 2n+3\quad \text{if n is odd.}\end{cases}$$
Prove that the range of $f$ is $\{m\in\mathbb{Z}|m\equiv 1,2,5,8 \ \text{or} \ 9 \pmod {12}\}$
I wasn't sure of a nice way to ... | Since they're asking for a result modulo $12$, I would've gone for a more direct approach, like so:
$(3)(4) = 12$, so for the even cases, work modulo $4$. An even number $n$ satisfies $n \equiv 0,2 \pmod 4$. So $3n+2 = 2,8 \pmod{12}$.
$(2)(6) = 12$, so for the odd cases, work modulo $6$. An odd number $m$ satisfies $m ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Convergence of $n \sqrt{n^2+1}-n^2$ I have been asked to study the convergence of the the serie $a_n = n\sqrt{n^2+1}-n^2$. I have been told that the answer is $\frac{1}{2}$ but I am not being able to demonstrate it. I have tried to integrate it and i get that $\int^\infty_1 n\sqrt{n^2+1}-n^2 = \frac{1}{3}[(n^2+1)^{3/2}... | Consider using
$$\sqrt{1 + x} = 1 + \frac{x}{2} - \frac{x^{2}}{8} + \mathcal{O}(x^{3})$$
for which
\begin{align}
f(n) &= n \, \sqrt{n^{2} + 1} - n^{2} = n^{2} \, \left(\sqrt{1 + \frac{1}{n^{2}}} - 1 \right) \\
&= n^{2} \, \left(\frac{1}{2 \, n^{2}} - \frac{1}{8 \, n^{4}} + \mathcal{O}\left(\frac{1}{n^{6}}\right) \right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2498902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Proof for an identity (from Ramanujan written) I saw an identity by Ramanujan
$$\forall n \in \mathbb{N} ,n>1 :\lfloor \sqrt n+\sqrt {n+2}+\sqrt{n+4} \rfloor=\lfloor \sqrt {9n
+17}\rfloor$$ I tried to prove it by limit definition . I post my trial below . If possible check my prove (right , wrong) ?
Then Is there more... | Assume $(m-1)^2<9n+17<m^2$. (We never get equality because $9n+17\equiv -1\pmod{3}$.) So we get: $$(m-1)^2+1\leq 9n+17\leq m^2-1$$
or $$\frac{(m-1)^2-16}{9}\leq n\leq\frac{m^2-18}{9}.$$
So $$\frac{\sqrt{(m-1)^2-16}}{3}\leq\sqrt{n}\leq\frac{\sqrt{m^2-18}}{3}\\
\frac{\sqrt{(m-1)^2+2}}3\leq\sqrt{n+2}\leq\frac m3\\
\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2501870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Find Probability that even numbered face occurs odd number of times A Die is tossed $2n+1$ times. Find Probability that even numbered face occurs odd number of times
First i assumed let $a,b,c,d,e,f$ be number of times $1$ occurred, $2$ occurred and so on $6$ occurred respectively. Then we have
$$a+b+c+d+e+f=2n+1$$ i... | First let us simplify the problem: the probability of an even face is 1/2, so is the probability of an odd face. The problem is therefore equivalent to a number of coin tosses of a fair coin, with even/odd faces corresponding to heads/tails.
Define the events $E$: the number of heads is even and $F$: the number of tail... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Computing and proving existence of integral We're supposed to prove the existence of the following integral and compute it.
$\int^{10}_{y=0}\int^{\pi/3}_{x=0}{xy \cos x y^2}\,\mathrm{d}x\mathrm{d}y$
My two cents on this. I was trying to use Fubini's theorem and change the order of integration, but ran into trouble eval... | Well, we know that:
$$\mathscr{I}:=\int_0^{10}\int_0^\frac{\pi}{3}x\cdot\text{y}\cdot\cos\left(x\cdot\text{y}^2\right)\space\text{d}x\space\text{d}\text{y}=$$
$$\int_0^{10}\text{y}\cdot\left\{\int_0^\frac{\pi}{3}x\cdot\cos\left(x\cdot\text{y}^2\right)\space\text{d}x\right\}\space\text{d}\text{y}\tag1$$
Using:
$$\cos\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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let $f(\frac{x}{3})+f(\frac{2}{x})=\frac{4}{x^2}+\frac{x^2}{9}-2$ then find $f(x)$ let $f(\frac{x}{3})+f(\frac{2}{x})=\frac{4}{x^2}+\frac{x^2}{9}-2$
then find $f(x)$
My Try :
$$f(\frac{x}{3})+f(\frac{2}{x})=(\frac{2}{x})^2-1+(\frac{x}{3})^2-1$$
So we have :
$$f(x)=x^2-1$$
it is right ?Is there another answer?
| Suppose $f(x)$ can be given by the power series expansion $f(x) = a_{0} + a_{1} x + a_{2} x^{2} + \cdots$. Now, for the equation
$$f\left(\frac{x}{3}\right) + f\left(\frac{2}{x}\right) = \frac{x^{2}}{9} - 2 + \frac{4}{x^{2}}$$
it is seen that:
\begin{align}
\frac{x^{2}}{9} + \frac{4}{x^{2}} - 2 &= 2 a_{0} + a_{1} \, \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Explicit sum of $\sum_{n=0}^{\infty} \frac{6+3^n}{6^{n+2}}$ I want to find the sum of the infinite series $$\sum_{n=0}^{\infty} \frac{6+3^n}{6^{n+2}}$$
So far I have managed to simplify the expression to $$\sum_{n=0}^{\infty} \left(\frac {1}{6^{n+1}} + \frac{1}{9\cdot2^{n+2}}\right)$$.
The sum clearly converges as both... | for the finite sum we have $$\sum_{n=0}^k\frac{6+3^n}{6^{n+2}}=-4/5\,{\frac {6+{3}^{k+1}}{{6}^{k+3}}}-{\frac {72+12\,{3}^{k+2}}{5\,{6
}^{k+4}}}+{\frac{23}{90}}
$$ now compute the limit for $k$ tends to infinity
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2503207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Limit of a sequence by definition I need to prove by definition that the limit of sequence :
$$a_n = \frac{5n^3-3n^2+1}{4n^3+n+2}$$
is $\dfrac54$ which means I need to show that :
$$\left|\frac{5n^3-3n^2+1}{4n^3+n+2} - \frac54\right| < \varepsilon$$
I tried for hours to solve it but could not,
Can anyone help please?
T... | $|f(n)|:=|\dfrac{5n^3-3n^2+1}{4n^3+n+2} -5/4| =$
$|\dfrac{-12n^2 -5n-6}{4(4n^3+n+2)}| \lt$
$|\dfrac{12n^2 +5n+6}{16n^3}| \lt$
$|\dfrac{12n^2+5n^2+6n^2}{n^3}| =$
$|\dfrac{23n^2}{n^3}| = |\dfrac{23}{n}|.$
Let $\epsilon >0$ be given:
Choose $M \gt 23/\epsilon,$ with $M$, real.
There is a $n_0 \gt M$.(Archimedes)
Then for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Can't solve integral of Landau Lifschitz Classical Mechanics I am reading right now a book about classical mechanics (Landau Lifschitz) which contains an integral which I do not understand
$\int_{0}^{\alpha}\left [ \frac{dx_{2}(U)}{dU}- \frac{dx_{1}(U)}{dU}\right]dU\int_{U}^{\alpha}\frac{dE}{\sqrt{(\alpha - E)(E-U)}}$
... | This is a $\int\frac{du}{\sqrt{a^2 - u^2}}$ form, which you can find in integral tables and which most relatively thorough elementary calculus texts discuss. For a very complete treatment of the kinds of classical integration methods that would be familiar to students of the Landau/Lifschitz era, see A Treatise on the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to calculate the $\gcd(10^6 +1 , 10^2 +1)$? I can't really figure out how to approach this question..
I have tried to factor 10^2 +1 out of 10^6 +1 , however, the '+1' part makes it difficult.
$10^6 + 1 = (10^2 +1) \cdot 10^3 + 899000$,
$10^2 + 1 = 899000 \cdot ...? $
Here is where I get stuck, because $899000$ is... | Well, the "throw rocks at it until it falls off a cliff approach":
$\gcd (10^6 + 1, 10^2 + 1) = \gcd([10^6 + 1] - 10^4(10^2 + 1), 10^2 + 1)$
$=\gcd (-10^4 + 1, 10^2 + 1) = \gcd(-10^4 + 1 + 10^2(10^2 + 1), 10^2 + 1)$
$= \gcd(10^2 + 1,10^2+1) = 10^2 + 1$.
....
There's also the insightful. $x^n - 1 = (x-1)(x^{n-1} + x^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2507510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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An expansion of $\pi \cot(\pi/k)/(k-2)$. Mathematica suggests that for $k>=3$,
$$
\sum _{r=0}^{\infty } \frac{\left(\frac{2}{k}\right)_r}{(r+1) \left(1+\frac{1}{k}\right)_r}
=
\frac{\pi \cot \left(\frac{\pi }{k}\right)}{k-2}
,
$$
where $(x)_r=x(x+1)\dots(x+r-1)$.
A quick look of DLMF does not find any expansion of $\c... | With Euler’s integral transform we get :
$$\begin{eqnarray} \sum\limits_{n=0}^\infty \frac{(2x)_n}{(n+1)(1+x)_n} & = & {_3 F_2}(1,1,2x;2,x+1;1) \\ & = &\frac{\Gamma(x+1)}{\Gamma(2x)\Gamma(1-x)} \int\limits_0^1 t^{2x-1} (1-t)^{-x} {_2 F_1}(1,1;2;t)dt \\ & = & \frac{\Gamma(x+1)}{\Gamma(2x)\Gamma(1-x)} \int\limits_0^1 t^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2507818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Hi, I need help with the assignment, Find the point $P$ on the sphere $x^2 + y^2 + z^2 − 8 x − 4 y − 10 z = 4$ which is furthest from the plane Find the point $P$ on the sphere
$x^2 + y^2 + z^2 − 8x − 4y − 10z = 4$
which is furthest from the plane
$6x + 2y − 9z = −23$.
| Let the point $P \equiv (p, q, r)$
The distance of $P$ from $6x+2y−9z=−23$ is given by $\frac{6p+2q−9r+23}{\sqrt{6^2+2^2+9^2}} = \frac{6p+2q−9r+23}{11}$
In other words, we are required to maximize $\frac{6p+2q−9r+23}{11}$ subject to $p^2+q^2+r^2−8p−4q−10r-4= 0$
This is equivalent to the following problem:
Maximize $6p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2508245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$. Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$.
I know there exists some slick technique to simplify this problem. Any hints are greatly appreciated.
| HINT
let $d=(2a^2+29a+65,a+13)$. We notice that $-13$ is a root of $2a^2+29a+39$. Another root is $-\frac32$
So we have $2a^2+29a+65=(a+13)(2a+3)+26$
Thus $d=((a+13)(2a+3)+26,a+13)\Rightarrow$ $d|(a+13)(2a+3)+26, d|a+13\Rightarrow d|(a+13)(2a+3)\Rightarrow d|26$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2513736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Prove that $\cosh(x)=\sec(\theta )$ if $x=\ln(\sec \theta + \tan \theta)$ I'm trying to prove that $\cosh(x)=\sec(\theta )$ if $x=x=\ln(\sec \theta + \tan \theta)$. I've substituted the value of $x$ into $\cosh(x)$ to get $$\frac{e^{\ln(\sec \theta + \tan \theta)}+e^{-\ln(\sec \theta + \tan \theta)}}{2}$$ and simplifie... | You're almost there!
$$\frac{\sec \theta + \tan \theta+\frac{1}{\sec \theta + \tan \theta}{}}{2}=\frac{1}{2}\left(\frac{1}{\cos \theta}+\frac{\sin \theta}{\cos \theta}+\frac{1}{\frac{1}{\cos \theta}+\frac{\sin \theta}{\cos \theta}}\right)$$ This simplifies to $$\frac{1}{2}\left(\frac{1+\sin\theta}{\cos \theta}+\frac{\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2514291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Find the number of real roots for $x+\sqrt{a^2-x^2}=b$, $a>0$, $b>0$, as a function of $a$ and $b$
Given: (1) $x+\sqrt{a^2-x^2}=b$, $(a,b,x)\subset \mathbb R$, $a>0$, $b>0$.
Find: number of roots for (1), given possible values for $a$ and $b$.
This is a question from a book for the preparation for math contests.
It s... | You just forgot to check those conditions you mentioned. Indeed, you should check that for what values of $a$ and $b$ the conditions $-a\le x\le a$ and $b-x \ge 0$ are satisfied. The difficulty is to consider a few if then conditions.
If $b=\sqrt{2}a$ then $x=\frac{b}{2}$ is a potential solution. We should make sure th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2514902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 1
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Prove directly from the definition that $({1\over2}+\frac{1}{2^2}+...+\frac{1}{2^n})_n $ is Cauchy Prove directly from the definition that $({1\over2}+\frac{1}{2^2}+...+\frac{1}{2^n})_n$ is cauchy
I know from the definition of Cauchy that |$x_n$-$x_m$|< ϵ but how do you do this with |$\frac{1}{2^n}- \frac{1}{2^m}$|
wha... | Let $S_n=\sum_{k=1}^n\frac{1}{2^k}$. Then, we have for any $\epsilon>0$
$$\begin{align}
\left|S_n-S_m\right|&=\left|\sum_{k=\min(m,n)+1}^{\max(m.n)}\frac{1}{2^k}\right|\\\\
&=\left|\frac{1}{2^{\min(n,m)}}-\frac{1}{2^{\max(m,n)}}\right|\\\\
&\le \frac{1}{2^{\min(n,m)-1}}\\\\
&<\epsilon
\end{align}$$
whenever $\min(n,m)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2515137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
What is $\gcd (x^{3}+6x^{2}+11x+6,x^{3}+1)$ When applying straight-forward Euclid's algorithm the result have fractional coefficients, but by factoring linear terms you get $x+1$. Which answer is right?
| $(x^3+6x^2+11x+6,x^3+1)=((x+1)(x^2-x+1),(x+1)(x+2)(x+3)) = (x+1)((x^2-x+1),(x+2)(x+3)) = (x+1)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Polynomial remainder when divisor has double roots How to find the remainder of the division of $P(x) = x^{444} + x^{111} + x - 1$ by $Q(x) = (x^2 + 1)^2$?
By the little Bézout's theorem, we can write
$$P(x) = (x^2 + 1)^2 W(x) + ax^3+bx^2+cx+d.$$
In fact, I've found out that, since $(x^2 + 1)$ divides $P(x)$, we can wr... | We have
\begin{eqnarray}
P(x)&=&(x^2+1)^2W(x)+ax^3+bx^2+cx+d\\
P'(x)&=&4x(x^2+1)W(x)+(x^2+1)^2W'(x)+3ax^2+2bx+c
\end{eqnarray}
and
$$
P(i)=i^{444}+i^{111}+i-1=(i^4)^{111}+i^{108}i^3+i-1=1-i+i-1=0
$$
It follows that
$$
0=P(i)=ai^3+bi^2+ci+d=-ai-b+ci+d=(d-b)+(c-a)i
$$
Therefore
$$
a=c,\quad b=d
$$
Also
$$
P'(x)=444x^{443... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2520110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Simplifying expressions. How do you simplify the following expression?
$$Q=(1-\tan^2(x)) \left(1-\tan^2 \left(\frac{x}{2}\right)\right)\cdots \left(1-\tan^2\left(\frac{x}{2^n}\right)\right)$$
I've tried the the following :
$\tan x = \frac{\sin x}{\cos x}$
$$\cos x = \frac{\sin 2x}{2\sin x}$$
$$\tan x = \frac{2\sin^2x}{... | A more straightforward hint comes from the tangent double angle formula:
$$
\tan(2x) = \frac{2\tan{x}}{1-\tan^2 x}.
$$
You'll then find you have a telescoping product.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2520516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Prove or Disprove There Exist Only One Pair of Four Positive Integers whose sum equals their product. I have the following proof:
"We know that $$1+2+3=1\cdot 2\cdot 3.$$ Prove or Disprove: There exist four positive integers whose sum equals their product."
Now this seems pretty obvious at first that it can never exist... | To complete the task, we shall prove that the only solution is $1,1,2,4$.
First, we shall prove the following:
Assume that $x,y$ are real numbers such that $y\ge x\ge 2$. Then
$xy\ge x+y$ and if $xy=x+y$ then $x=y=2$.
Proof:
$xy\ge 2y=y+y\ge x+y$.
If $x>2$ then $xy>2y\ge x+y$.
If $y>x$ then $xy\ge 2y>x+y$.
N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2521422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Maximum possible value of $P(10)$ The real numbers $a$, $b$, $c$, and $d$ are each less than or equal to $12$. The polynomial $$P(x)=ax^3+bx^2+cx+d$$ satisfies $P(2)=2$, $P(4)=4$ and $P(6)=6$. Find the maximum possible value of $P(10)$.
What I did was first I used the given information to get $3$ equation in $a, b,c,d... | From the system
$
\left\{
\begin{array}{l}
8 a+4 b+2 c+d=2 \\
64 a+16 b+4 c+d=4 \\
216 a+36 b+6 c+d=6 \\
\end{array}
\right.
$
we get
$b = -12 a, c = 1 + 44 a, d = -48 a$
with the constraints
$-12 a\leq 12\land 44 a+1\leq 12\land -48 a\leq 12$
which give the limitation
$-\frac{1}{4}\leq a\leq \frac{1}{4}$
Therefore... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2521862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
What is the number of 5 digit numbers divisible by 3? What is the number of $5$ digit numbers divisible by $3$ using the digits
$0,1,2,3,4,6,7$ and repetition is not allowed?
| I mostly agree with the previous answer. You can add up all seven numbers in that set which is equal $23$. For a five digit number to be divisible by $3$ the sum of its digits has to be a multiple of three.
So we have to find all possible combinations of $5$ digit numbers that when all digits are added equals a multipl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2522608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Proof $\lim\limits_{n \rightarrow \infty} \sqrt{n} (3^{\frac{1}{n}} - 2^{\frac{1}{n}}) = 0$ How would one go about proving the limit of sequence:
$$\lim_{n \rightarrow \infty}\ \sqrt{n} (3^{\frac{1}{n}} - 2^{\frac{1}{n}}) = 0\,. $$
Epsilon definiton of limit seems to be too complicated and/or unsolvable (variable both... | From
$$1=3-2=\left(3^{\frac{1}{n}}\right)^n-\left(2^{\frac{1}{n}}\right)^n=\\
\left(3^{\frac{1}{n}}-2^{\frac{1}{n}}\right)\left( 3^{\frac{n-1}{n}}+3^{\frac{n-2}{n}} 2^{\frac{1}{n}}+3^{\frac{n-3}{n}} 2^{\frac{2}{n}}+...+3^{\frac{1}{n}} 2^{\frac{n-2}{n}} + 2^{\frac{n-1}{n}}\right)$$
we have
$$0<\sqrt{n}\left(3^{\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2523876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
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GCD of two elements in $\mathbb Z \left[\frac{1 + \sqrt{-11}}{2}\right]$ I have to find $(3 + \sqrt{-11}, 2 + 4\sqrt{-11})$ in $\mathbb Z \left[\frac{1 + \sqrt{-11}}{2}\right]$.
If $\mathbb Z \left[\frac{1 + \sqrt{-11}}{2}\right]$ is an Euclidean domain, the euclidean algorithm should be acceptable for computing the GC... | Well, let's see: $$\frac{3 + \sqrt{-11}}{2 + 4 \sqrt{-11}} = \frac{5 - 11 \sqrt{-11}}{18}.$$ Since $$\frac{5}{18} \approx \frac{1}{2}$$ and $$\frac{-11}{18} \approx -\frac{1}{2}$$ we have $$3 + \sqrt{-11} = (2 + 4 \sqrt{-11})\left(\frac{1}{2} - \frac{\sqrt{-11}}{2}\right) - 20.$$
Yeah, that is a problem. Try $$\frac{5 ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Solving system of linear equations with rationals Problem:$$(x+11):(x+6)=(y+12):(y+7)\land(y+1):(x-1)=y:x$$
When I tried using $\frac{a1}{b1}=\frac{a2}{b2}=\frac{k1a1+k2a2}{k1b1+k2b2}$ for $\frac{x+11}{x+6}=\frac{y+12}{y+7}$ where k1 = 1 and k2 = -1, this is what I got:$$\frac{x+11-y-12}{x+6-y-7}=\frac{x-y-1}{x-y-1}=1$... | Hint:
First absurd result came up as $x-y-1$ is actually $=0$ here.
For example $$\dfrac23=\dfrac46=\dfrac{2\cdot2-4}{2\cdot3-6}=?$$
Just cross multiply to form two simultaneous equations in $x,y$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2526489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate line integral $\int_c x\,dx+ y\,dy + z\,dz$, where $C$ is the straight line from $(1,0,0)$ to $(0,1,\pi/2)$
$ \displaystyle \int_c x\,dx+ y\,dy + z\,dz$, where $C$ is the straight line from
$(1,0,0)$ to $(0,1,\pi/2)$
Parametric form of line will be:
$$x=1 - t, \quad y= t, \quad z= \frac{\pi t} 2$$
The inte... | Given $x = 1-t, y = t, z = \frac{\pi}{2} t$ then,
\begin{align*} x\, \textrm{dx} + y\, \textrm{dy} + z\, \textrm{dz} &=(1-t)(-1)\,dt + t(1)\, \rm{dt} + \frac{\pi}{2} t \left( \frac{\pi}{2}\right) \, \rm{dt} \\ \\ & = \left(-1+t+t+ \frac{\pi^2}{4}t\right) \, \rm{dt} \\ \\ & = \left( t \left( 2+\frac{\pi^2}{4} \right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2530946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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writing Cosines using De Moivre's formula Given the question:
Use De Moivre’s formula to find a formula for $\cos(3x)$ and $\cos(4x)$ in terms of $\cos(x)$ and $\sin(x)$. Then use the identity $\cos^2(x) + \sin^2(x) = 1$ to express these formulas only in terms of $\cos(x)$.
I started out by rewriting $\cos(3x)$:
$\co... | $${ \left( \cos { x } +i\sin { x } \right) }^{ 3 }=\cos { \left( 3x \right) +i\sin { \left( 3x \right) } } \\ \cos ^{ 3 }{ x } +3i\cos ^{ 2 }{ x\sin { x } -3\cos { x\sin ^{ 2 }{ x } -i\sin ^{ 3 }{ x } = } } \cos { \left( 3x \right) +i\sin { \left( 3x \right) } } \\ \\ \cos { \left( 3x \right) =\cos ^{ 3 }{ x } -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find all pairs of complex numbers (x,y) satisfying $xy=-1$ and $x^3-y^3=-1$ the problem asks me to find all pairs of complex numbers (x,y) satisfying $xy=-1$ and $x^3-y^3=-1$ simultaneously. I set the two equations to equal each other which yields
$x^3-xy-y^3=0$ but from this point on on im stuck on how i would solve f... | It seems that anon gave some explanation of the suggestion I made.
Also, you do have questions using complex numbers, including some with $e^{m \pi i / n}.$
So, your original variable $x$ satisfies $x^6 + x^3 + 1 = 0.$ Note that we have $x^3 \neq 1,$ otherwise we would have $1+1+1=0,$ which is false. Next,
$$ 0 = (x^3 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2533217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Can we generalize $\sum_{n=0}^\infty \binom{4n}{n}\frac{1}{(3n+1)\,2^{4n+1}}=\frac1{T}$ for tribonacci constant $T$? (Inspired by this post.) Given the tribonacci constant $\Phi_3$, the tetranacci constant $\Phi_4$, etc. How do prove that,
$$\sum_{n=0}^\infty \binom{4n}{n}\frac{1}{(3n+1)\,2^{4n+1}}=\frac{1}{2}{\;}_3F_2... | Note that $\frac{1}{kn+1}\binom{(k+1)n}{n}$ is the number of $(k+1)$-ary trees on $n$ vertices. Let $T_{k+1}$ be the ordinary generating function for the class of $(k+1)$-ary trees. Then $T_{k+1}$ satisfies the equation
$$
T_{k+1}(z)=1+zT_{k+1}^{k+1}(z).
$$
Let $a=\frac{1}{2}T_{k+1}\left(\frac{1}{2^{k+1}}\right)$, then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2533893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Determine if the series 1/5 + 1/8 + 1/11 + 1/14 + 1/17 ... is convergent or divergent I am trying to determine if the series $\frac{1}{5} + \frac{1}{8} + \frac{1}{11} + \frac{1}{14} + \frac{1}{17} $ is convergent or divergent?
I can see that the terms in the denominator differ by 3 each time.
| The pattern for the denominator is found by first adding another term so that the series starts $\frac{1}{2} + \frac{1}{5} + \frac{1}{8} + ...$ then for the series $2 + 5 + 8 + 11 ... $ the general nth term is found as $ T_n = a + (n-1)d$ where $a = \frac{1}{2}$ is the first term and $d = 3$ the difference. So $T_n = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2535390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Representing a given number as the sum of two squares.
There are four essentially different representations of $1885$ as the sum of squares of two positive integers. Find all of them.
I am guessing the must be some nice way of solving $z= x^2 + y^2 $ where x and y are integers in general that i don't know.
Is ther... | Since $\mathbb{Z}[i]$ is a UFD, by letting
$$ r_2(n)=\left|\left\{(a,b)\in\mathbb{Z}^2:a^2+b^2=n\right\}\right|$$
we have
$$ r_2(n) = 4(\chi_4*1)(n) = 4\sum_{d\mid n}\chi_4(d) $$
where $\chi_4(m)$ equals $1$ if $m\equiv 1\pmod{4}$, $-1$ if $m\equiv 3\pmod{4}$ and zero otherwise.
This is essentially due to the Brahmagu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2536097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Trigonometric equation result differs from given i've got an equation:
$$\sin^6(x) + \cos^6(x) = 0.25$$
and i'm trying to solve it using the sum of cubes formula, like this:
$$ (\sin^2(x))^3 + (\cos^2(x))^3 = 0.25 $$
$$ (\sin^2(x) + \cos^2(x))^2 (\sin^4(x) - \sin^2(x)\cos^2(x) + \cos^4(x)) = 0.25 $$
$$ 1 - 3\sin^2(x)\c... | Both the answers are correct. Since, both represent the same solution set.
This is so because each odd number can be either represented as $2n+1$ and $2m-1$, or represented as $4p+1$ or $4q-1$. And hence $$2x=(2n\pm 1)\frac{\pi}{2} \implies \frac{n\pi}{2} \pm \frac{\pi}{4} $$
$$ \text{also} \; 2x =(4n\pm 1)\frac{\pi}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculate Riemann sums for $f(x)=4+3x^3$ for $x \in [-2,1]$ I have the following exercise:
Let $f(x)$ = $4+3x^3$, $x$ $\in$ $[-2; 1]$. Calculate the Riemann sums for $f$ on $[-2; 1]$ by dividing the interval into n equal sub-intervals and taking midpoints as sample points. Find the limit of Riemann sums as $n \to \inft... | Mistake $1$:
$$\lim_{n \to \infty}\frac{1}{\color{blue}n}\sum_{k=1}^{n}\left[4+\color{red}3(\frac{6k-3-4n}{2n})^3\right]= \color{red}{4}+
\lim_{n \to \infty}\frac{3}{\color{red}{8n^4}}\sum_{k=1}^{n}(6k-(3+4n))^3$$
Mistake $2$:
\begin{align}&\lim_{n \to \infty}\frac{3}{2n^3}\sum_{k=1}^{n}(216k^3-108k^2(3+4n)+18k(3+4n)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $p$ is prime, $p\ne3$ then $p^2+2$ is composite I'm trying to prove that if $p$ is prime,$p\ne3$ then $p^2+2$ is composite. Here's my attempt:
Every number $p$ can be put in the form $3k+r, 0\le r \lt 3$, with $k$ an integer. When $r=0$, the number is a multiple of 3, so that leaves us with the forms $3k+1$ and $3k+... | Another observation is that primes $p\geq 5$ (the remaining $p=2$ can be checked manually) are of the following two forms $6k+1$ or $6k+5$ (simply because $6k+2, 6k+3, 6k+4$ are never primes). Then
*
*$p^2+2=36k^2+12k+3$ is divisible by 3
*$p^2+2=36k^2+60k+27$ is divisible by 3
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Is there a positive sequence $(a_n)_{n=1}^\infty$ where both $\sum_{n=1}^\infty \frac{1}{a_n}$ and $\sum_{n=1}^\infty \frac{a_n}{n^2}$ converge? Suppose $a_n$ is a positive sequence but not necessarily monotonic.
For the series $\sum_{n=1}^\infty \frac{1}{a_n}$ and $\sum_{n=1}^\infty \frac{a_n}{n^2}$ I can find exampl... | Both series cannot converge.
Suppose $\sum a_n/n^2$ converges. Since the divergent harmonic series can be written as
$$\sum_{n=1}^\infty \frac{1}{n} = \sum_{\frac{1}{n} \leqslant \frac{a_n}{n^2}} \frac{1}{n} + \sum_{\frac{1}{n} > \frac{a_n}{n^2}} \frac{1}{n},$$
and the first series on the RHS converges, it follows tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2545226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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On the way of solving Integral $\int_{0}^{2\pi}\frac{x\sin^{2n}x}{\sin^{2n}x+\cos^{2n}x}dx$. Question
For $n > 0$, Find
$$\int_{0}^{2\pi}\frac{x\sin^{2n}x}{\sin^{2n}x+\cos^{2n}x}dx$$
My Approach
Let
$I_{n}$ =$$\int_{0}^{2\pi}\frac{x \sin^{2n}x}{\sin^{2n}x+\cos^{2n}x}dx$$
I$_{n}$=$\int_{0}^{2\pi}$$\frac{\left(2\p... | I will assume $n \in \mathbb{N}$. If in the integral
$$I_n = \int^{2\pi}_0 \frac{x \sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx,$$
we set $x \mapsto 2\pi - x$ it can be readily seen that
$$I_n = \pi \int^{2\pi}_0 \frac{\sin^{2n} x}{\sin^{2n} x + \cos^{2n} x} \, dx.$$
Note the integrand
$$f(x) = \frac{\sin^{2n} x}{\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving linear congruent system
Consider the following congruences system:
$$3x \equiv 1 \pmod{8} \\ x \equiv 7 \pmod {12} \\ x \equiv 4 \pmod
{15}$$
Find a minimal solution for the system
I was able to show using the Euclidean algorithm that $(-5)\cdot 3\equiv 1 \pmod 8$. so every solution for the first congruence... | HINT reduce the system to $\pmod {p_i}$ and apply CRT
You can expand in this way:
$\begin{cases} 3x \equiv 1 \pmod{2^3} \\ x \equiv 7 \pmod {2^2\cdot3} \\ x \equiv 4 \pmod{3\cdot5} \end{cases}$
$\begin{cases} x \equiv 3 \pmod{2^3} \\ x \equiv 3 \pmod {2^2} \\x \equiv 1 \pmod {3} \\ x \equiv 4 \pmod{5} \end{cases}$
$\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $2^{3n+1} + 5$ is a multiple of 7 for all n ≥ 0. As the title states I need to prove that $(2^{3n+1}+5)$ is a multiple of 7 for all $n \geq 0$.
I can do this using induction but I also want to prove it using modular arithmetic. So here's what I've got so far.
Start with Fermat's little theorem:
\begin{al... | Here's a much quicker way to do it:
Notice that $2^{3n+1} = 2^{3n} \cdot 2 = 8^n \cdot 2$. Since $8 \equiv 1 \pmod 7$, we have $1 * 2 + 5 \equiv 0 \pmod 7$, which is clearly true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Prove $5^n + 2 \cdot 3^n - 3$ is divisible by 8 $\forall n\in \mathbb{N}$ (using induction)
Prove $5^n + 2 \cdot 3^n - 3$ is divisible by 8 $\forall n\in \mathbb{N}$
Base case $n = 1\to 5 + 6 - 3 = 8 \to 8 \mid 8 $
Assume that for some $n \in \mathbb{N}\to 8 \mid 5^n + 2 \cdot 3^n - 3$
Showing $8 \mid 5^{n+1} + 2 ... | $$4 \cdot 5^n+4 \cdot 3^n$$ is clearly divisible by 8 cause once you take the 4 out it is the sum of two odd numbers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2554949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
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How to calculate this hard integral $\int_0^{\infty} \frac{\arctan(x)\sqrt{\sqrt{1+x^2}-1}}{x\sqrt{1+x^2}}\,dx$? How to prove that $\displaystyle \int_0^{\infty} \dfrac{\arctan(x)\sqrt{\sqrt{1+x^2}-1}}{x\sqrt{1+x^2}}\,dx= \frac{1}{4}\pi^2\sqrt{2}-\sqrt{2}\ln^2\left(1+\sqrt{2}\right)$ ?
It's very difficult and I have no... | $\begin{align}I&=\int_0^{\infty} \dfrac{\arctan(x)\sqrt{\sqrt{1+x^2}-1}}{x\sqrt{1+x^2}}\,dx
\end{align}$
Perform the change of variable $x=\dfrac{2y}{1-y^2}$,
$\begin{align}I&=\sqrt{2}\int_0^{1} \dfrac{\arctan\left(\dfrac{2y}{1-y^2}\right)}{\sqrt{1-y^2}}\,dy\\
&=2\sqrt{2}\int_0^{1} \dfrac{\arctan y}{\sqrt{1-y^2}}\,dy\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2557405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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How many possibilities are there for two full houses to be dealt to two players in one game? Imagine dealing cards from a classic 52 card deck to two poker players.
How many possibilities are there for both of them to be dealt a full house(three cards in same rank and two cards of another rank) in same round?
As I kno... | *
*choose the two three-card ranks. The number of ways is $\binom{13}{2}\binom{4}{3}\binom{4}{3}$.
*choose a two-card rank. The number of ways is $\binom{11}{1}\binom{4}{2}$.
*the next two-card rank may be the same or different as the first one. The number of ways is $\binom{2}{2}+\binom{10}{1}\binom{4}{2}$
*Take ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2558693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove through induction that if $f(x) := \frac{2}{1-x^2}$ then $f^{(n)}(x) = n!(\frac{1}{(1-x)^{n+1}}+\frac{(-1)^n}{(1+x)^{n+1}})$ The original formula is $f(x) := \frac{2}{1-x^2}$, for $f\colon\mathbb{R}- ({-1,1}) \to \mathbb{R}$
Through Induction prove that $f^{(n)}(x)=n!(\frac{1}{(1-x)^{n+1}}+\frac{(-1)^n}{(1+x)^{n+... | you did all right until the last line
$$....=(n!)\left[(\frac{1}{(1-x)^{n+1}}\right]' + \left[\frac{(-1)^n}{(1+x)^{n+1}}\right]' =(n!)(\frac{n+1}{(1-x)^{n+2}} - \frac{(n+1)(-1)^n}{(1+x)^{n+2}} \\= (n+1)!\left(\frac{1}{(1-x)^{n+2}} + \frac{((-1)^{n+1}}{(1+x)^{n+2}}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2563663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Time and Work. How much work does C do per hour?
A, B and C need a certain unique time to do a certain work. C needs 1
hours less than A to complete the work. Working together, they require
30 minutes to complete 50% of the job. The work also gets completed if
A and B start working together and A leaves after 1 ... | Let us assume, they take $a, b,$ and $c$ hours to finish the job individually, respectively. Note the following points:
*
*They complete half the job in half an hour $\implies$ full job is completed in a hour. Hence, $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$
*Also, A takes 1 hour more time than C. So, $a= c+1\implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2566277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Probability Problem with $10$ players being put on two teams Could somebody please check my answer to this problem?
Thanks
Bob
Problem:
$10$ kids are randomly grouped into an A team with five kids and a B team with five kids. Each
grouping is equally likely. There are three kids in the group, Alex and his two best frie... | The probability that Alex ends up on the same team as at least one of his two best friends can be found by subtracting the probability that neither of his friends is on the same team from $1$. If Alex is on a team, four of the other nine kids must be his teammates. They can be selected in
$$\binom{9}{4}$$
ways. If ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\frac{AQ* RB}{QR}$ is constant, while point $P$ moves along the ray $AB$. Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$. Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$. Prove that $\frac{AQ* RB}{QR}$ is constant, while po... |
Let $|AT|=a$, $|QT|=b$, $|QB|=c$,
$|AP|=u$, $|BP|=v$,
$|TP|=p$, $|QP|=q$,
$|OA|=|OB|=|OP|=|OD|=R$ (circumradius of
$\triangle ABP$,
$\triangle ADP$,
$\triangle DFP$,
$\triangle FBP$).
Prove that
\begin{align}
\frac{(a+b)(b+c)}{b}
&=\mathrm{const},
\tag{1}\label{1}
\quad\text{ while point $P$ moves along the arc $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Transform $\iint_D \sin(x^2+y^2)~dA$ to polar coordinates and evaluate the polar integral.
Transform the given integral in Cartesian coordinates to polar coordinates and evaluate the polar integral:
(b) $\iint_D \sin(x^2+y^2)~dA$, where $D$ is the region in the first quadrant bounded by the lines $x=0$ and $y=\sqrt{3}... | Given the above integral with the given bounds converted to polar coordinates, $$\iint_D \sin(x^2+y^2)~dA=\int_{\pi/3}^{\pi/2} \int_{\sqrt{2\pi}}^{\sqrt{3\pi}} \sin(r^2)\cdot r~dr~d\theta$$ evaluated gives ${\pi/6}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Number of teams There are $8$ women and $7$ men, from which we must create a team of $4$ women and $3$ men. Two men doesn't like each other, so they don't want to be in one team.
I tried number of teams (without the specific two deviant men): $\binom{8}{4}\times \binom{7}{3}$ but it's not correct.
The given solution ... | Some thoughts:
1) No matter what, the number of women on the team can be computed as $\binom{8}{4}$
2) $\binom{8}{4}\times \binom{7}{3} = 2450$. $\binom{8}{4}\times \binom{6}{2}+\binom{8}{4}\times \binom{5}{3} = 1750$, not $2800$.
3) Count the teams containing neither of the special men. Then add the count of teams con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Finding smallest possible value of expression with x and y I'm not supposed to use calculus here. I'm trying to find the smallest possible value of the expression $x^2+4xy+5y^2-4x-6y+7$ for real numbers $x$ and $y$.
Here's my attempt:
$x^2+4xy+5y^2-4x-6y+7=[(x-2)^2+3)+5((y-3/5)^2-9/25)]+4xy$
$3\le (x-2)^2+3$ for all re... | minimal trickery, I see $u = x + 2y,$ so $4u = 4x+8y,$ your expression becomes
$$ u^2 + y^2 - 4u + 2y + 7, $$
$$ u^2 - 4 u + y^2 + 2 y + 7, $$
$$ (u-2)^2 - 4 + (y+1)^2 - 1 + 7, $$
$$ (u-2)^2 + (y+1)^2 + 2 $$
least value, $2,$ occurs when $y=-1$ and $x+2y = x - 2 = 2,$ so $x=4$ and $y=-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Slope of The Tangent to $r=7\sin \theta$ , $\theta = \frac{\pi}{6}$ Find slope of tangent to $r=7\sin \theta$ , $\theta = \frac{\pi}{6}$
using $$\frac{dy}{dx} = \frac{\frac{dr}{dθ}\sin \theta + r \cos \theta}{\frac{dr}{dθ}\cos \theta-r \sin \theta}$$
I got
$$\frac {14\cos \theta \sin \theta}{7 \cos^2 \theta-7\sin \thet... | When you have the equation in rectangular coordinates
$$
x^2+y^2=7y
$$
you can use implicit differentiation:
$$
2x+2yy'=7y'
$$
so
$$
y'=\frac{2x}{7-2y}
$$
For $\theta=\pi/6$, we have $r=7/2$, so $x=7\sqrt{3}/4$ and $y=7/4$; therefore the derivative at the point is
$$
y'(7\sqrt{3}/4)=\frac{2\dfrac{7\sqrt{3}}{4}}{7-2\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Why is function domain of fractions inside radicals not defined for lower values than those found by searching for domain of denominator in fraction? Consider function $y = \sqrt{\frac{1-2x}{2x+3}}$. To find the domain of this function we first find the domain of denominator in fraction:
$2x+3 \neq 0$
$2x \neq -3$
$x \... | $\frac ab > 0$ means either 1) !!BOTH!! $a > 0; b>0$ OR 2) !!BOTH!! $a < 0; b< 0$.
And as $\frac ab$ existing means $b \ne 0$ then $\frac ab > 0$ means either 1) both $a \ge 0; b < 0$ or both $a \le 0; b < 0$.
So $\frac{1-2x}{2x+3} \ge 0$ means
1) $1 - 2x \ge 0;2x + 3 > 0$
2) $1-2x \le 0; 2x + 3 < 0$.
Number 1) yeilds... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 0
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Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.