Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Find $\lim_{x\to 0} \cos \big(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \big) $ Find the limit $$\lim_{x\to 0} \cos \bigg(\pi x^2 \csc (\frac {x} {2}) \cot (6x) \bigg)$$
I dont even know where to get started... Some hints and solutions would be appreciated! Thanks in advance!
P.S typed this on an iphone, sorry for any mi... | Note that
$$\pi x^2 \csc (\frac {x} {2}) \cot (6x)=\frac{\pi}{3}\cdot\frac{\frac{x}{2}}{\sin(\frac{x}{2})}\cdot\frac{6x}{\sin(6x)}\cos(6x).$$
Since $\lim_{x\to 0}\frac{\sin x}{x}=1$ and $\lim_{x\to 0}\cos(x)=1$, we have
$$\lim_{x\to 0}x^2 \csc (\frac {x} {2}) \cot (6x)=\frac{1}{3}.$$
Since $\cos$ is a continuous func... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/203260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Is this composition of polynomial correct? Suppose $p(x)=x^2+5x+3$ and $q(x)=3x^3-3x+7$. Write the expression $(q \circ p)(x)$ as a sum of terms, each which is a constant times the power of $x$.
Here is my work for the problem:
$(q\circ p)(x)=3(x^2+5x+3)^3-3(x^2+5x+3)+7$
$(q\circ p)(x)=3(x^6+125x^3+27)-3x^2-15x-9+7$
$(... | As Jasper noted you did wrong here: $$...=3(x^2+5x+3)^3...$$ $$...=3(x^6+125x^3+27)-...$$ In fact $$(a+b+c)^3=a^3 + 3a^2b + 3a^2c + 3ab^2 + 6abc + 3ac^2 + b^3 + 3b^2c + 3bc^2 + c^3$$ we have $q(p(x))=3x^6+45x^5+252x^4+645x^3+753x^2+390x+79$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/203626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How many triangles I saw this question today, it asks how many triangles are in this picture.
I don't know how to solve this (without counting directly), though I guess it has something to do with some recurrence.
How can count the number of all triangles in the picture ?
| Say that instead of four triangles along each edge we have $n$. First count the triangles that point up. This is easy to do if you count them by top vertex. Each vertex in the picture is the top of one triangle for every horizontal grid line below it. Thus, the topmost vertex, which has $n$ horizontal gridlines below i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/203873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
"answer_count": 7,
"answer_id": 1
} |
Prove or disprove: For all positive integers $ n$ , $\sqrt[3]{n}+\sqrt[3]{n+1}$ are irrational numbers. Prove or disprove: For all positive integers $ n$ ,
$\sqrt[3]{n}+\sqrt[3]{n+1}$ are irrational numbers.
| Note that $(x+y)^{3} = x^{3} +y^{3} + 3xy(x+y).$ If $n$ or $n+1$ is the cube of an integer, the result is clear by the uniqueness of prime factorization, so we assume that neither is the cube of an integer. Then neither is $n(n+1)$. Set $x = n^{\frac{1}{3}}$ and $y = (n+1)^{\frac{1}{3}}.$ If $x+y$ is rational, then so ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How does $\zeta(1 - s)$ become $(-1/s + \cdots)$? Why is
$$\zeta(1 - s) = -\frac{1}{s} + \cdots$$
for small negative values of $s$?
A detailed explanation would be appreciated.
| For $s < 0$
$$ \begin{eqnarray}
\zeta(1-s) &=& \sum_{n=1}^\infty n^{s-1} = \int_1^\infty x^{s-1} \mathrm{d}x + \sum_{n=1}^\infty \left( n^{s-1} - \int_{n}^{n+1} x^{s-1} \mathrm{d}x \right) \\ &=& -\frac{1}{s} + \sum_{n=1}^\infty \left( n^{s-1} - \frac{(n+1)^s - n^s}{s} \right)
\end{eqnarray}
$$
Now, observe that th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to find the oblique asymptote of root of a function? In a test example I'm solving, the question asks to find the oblique asymptote of the following function:
$f(x) = \sqrt{4x^2+x+6}$
$x$ at $+\infty$
We have only learned how to do so with rational functions. Is there any general way of finding the oblique asympto... | By completing the square we get $f(x) = \sqrt{ (2x + 1/4)^2 + 95/16}$.
This means (after squaring both sides and taking $(2x + 1/4)^2$ to the left hand side and factoring) that $$( f(x) - (2x + 1/4) ) ( f(x) + (2x + 1/4) ) = 95/16$$ and hence $$f(x) - (2x + 1/4) = \frac{95/16}{f(x) + (2x + 1/4)}.$$
But $f(x) + (2x +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/205587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Double module in a inequality Can somebody explain to me (or link me a site which does) how to solve this?
$$
||x+1| -1| \geq 3
$$
I have no idea how to work out this double absolute value sign.
| Here is a solution with squaring technique
$$
\begin{align}
||x+1|-1|\geq 3&\Longleftrightarrow (|x+1|-1)^2\geq 3^2\quad\text{ since both sides of inequality are non negative}\\
&\Longleftrightarrow (|x+1|-1)^2-3^2\geq 0\\
&\Longleftrightarrow ((|x+1|-1-3)((|x+1|-1)+3)\geq 0\quad\text{ since}\quad a^2-b^2=(a-b)(a+b)\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Using generating functions find the sum $1^3 + 2^3 + 3^3 +\dotsb+ n^3$ I am quite new to generating functions concept and I am really finding it difficult to know how to approach problems like this. I need to find the sum of $1^3 + 2^3 + 3^3 +\dotsb+ n^3$ using generating functions. How do I proceed about it?
| You can do this more prettily with exponential generating functions. Note that
$$e^{kx} = \sum_n \frac{k^n x^n}{n!}$$
so
$$1+e^x+e^{2x} + e^{3x} + \cdots + e^{kx} = \sum_n \frac{(1+2^n+3^n + \cdots + k^n) x^n}{n!}.$$
The left hand side is
$$(e^{kx}-1) \cdot \frac{1}{e^x-1} = \left( kx + \frac{k^2 x^2}{2} + \frac{k^3 x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/209268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 0
} |
Common tangent to two circles
Find the equations of the common tangents to the 2 circles:
$$(x - 2)^2 + y^2 = 9$$
and
$$(x - 5)^2 + (y - 4)^2 = 4.$$
I've tried to set the equation to be $y = ax+b$, substitute this into the 2 equations and set the discriminant to zero, we then get a simultaneous quadratic equation... | The parametric equation of $(x-a)^2+(y-b)^2=r^2$ is $(x=a+r\cos C,y=b+r\sin C)$
Let $$\frac{y-(b+r\sin C)}{x-(a+r\cos C)}=m$$ be a tangent at $(a+r\cos C,b+r\sin C)$, then the distance of line from the center is equal to the radius.
$$r=\frac{\mid m(a-a-r\cos C)-b+b+r\sin C\mid}{\sqrt{m^2+1}}=\frac{r\mid\sin C-m\cos C\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/211538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 2
} |
Circle Least Squares Fit So my question is this:
Find the equation of the circle that gives the best least squares circle fit to the points $(-1,-2), (0,2.4), (1.1,-4),$ and $(2.4,-1.6).$
So far I have this general equation:
$2xc_1+2yc_2+(r^2-c_1^2-c_2^2)=x^2+y^2$ where $r^2-c_1^2-c_2^2 = c_3$
So then I think I creat... | It is true that the given equation $Ac = B$ has no solution. That is why we are trying to get an approximate answer (least-squares fit).
An explanation on Khan Academy
will give you an understanding of the result after approximating it.
If we cannot find the solution to $Ac = B$, we find solution to $A^TAc=A^TB$.
This... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/214661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Frequency of a trigonometric function - Where is my mistake? I need to find the frequency of the following trigonometric function.$$y=\sin^4(x)+\cos^4(x)$$
The "answers" section says the answer is: $$F_y=\frac{\pi}{2}$$
This is what i did:
Finding $\sin(x)^4$ frequency (I'll call it F1):
$$\cos(2x)=1-\sin^2(x)$$
$$\sin... | You have proved that $\pi$ is a period, but you have not shown that it is the smallest period.
I would tackle the problem in more or less the same way that you did, using double angle identities, but the algebra can be simplified. Note that
$$1=(\cos^2 x+\sin^2 x)^2=\cos^4 x+\sin^4 x +2\cos^2 x\sin^2 x.$$
It follows t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/215150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
How to solve this equation:$\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-5}}}}=5$ Please, help me to solve this equation:
$$\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-5}}}}=5$$
I think if we continue to square, it would have to face a high-order equation.
I guess $x = 30$, but do not know how to solve.
| Observe that $\pm \sqrt{[17,33] + [-7,7]} = \pm [\sqrt{10}, \sqrt{40}] \subset [-7;7]$
This implies that for $x \in [17,33]$, all the quantities $f_1(x), \ldots, f_8(x) = \sqrt{x \pm \sqrt {x \pm \sqrt {x \pm \sqrt {x - 5}}}}$ are well-defined.
Furthermore, for $x=17$, $\sqrt{17 + [-7,7]} = [\sqrt{10}, \sqrt{24}] < 5$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/216179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 4
} |
How to show that $\left(\frac{y}{y-1}\right)^y = \left( 1 + \frac 1{y-1}\right)^y $ I need to show
\[\left(\frac{y}{y-1}\right)^y = \left( 1 + \frac 1{y-1}\right)^y \]
I cannot understand how they made this step! Can someone explain how this works?
| We have
\[
\frac y{y-1} = \frac{y-1+1}{y-1} = \frac{y-1}{y-1} + \frac 1{y-1} = 1 + \frac 1{y-1}. \]
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/216944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Two interesting and basic integrals. Show: $$\int_0^\pi \sqrt{1 + \sin(x)}\,dx.\ = 4$$
and $$\int_0^\pi \frac{xdx}{1 + \cos^2(x)} = \frac{\pi^2}{2\sqrt{2}}$$
| For the first one, you can use the identity $\sin(x)=-\cos(\frac{\pi}{2}+x)$.
Then by half angle formula
$$1+\sin(x)=1- \cos(\frac{\pi}{2}+x)=2 \sin^2(\frac{\pi}{4}+\frac{x}{2})$$
Thus
$$\int_0^\pi \sqrt{1 + \sin(x)}\,dx.\ = \int_0^\pi \sqrt{2}\sin(\frac{\pi}{4}+\frac{x}{2})\,dx=-2\sqrt{2}\cos(\frac{\pi}{4}+\frac{x}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/217801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
How to eliminate these extra solutions? (finding the gcd of two expressions)
Prove that for any integer $n$, $\gcd (3n^2+5n+7, n^2+1)=1$ or $41$.
The following answer is convoluted because I've intentionally created excess solutions. However, I can't figure out how to eliminate them! Anyone?
Let $$d=\gcd (3n^2+5n+7, ... | From your last step, we get that $d = 1,3,41,123$.
Recall that $$n^2 \equiv 0,1 \pmod{3} \text{ (Why?)}$$ Hence, $3$ (or) $123$ does not divide $n^2+1$.
EDIT
Note that any $n$ is either $0 \pmod{3}$ or $\pm1 \pmod{3}$.
Hence, $n^2 \equiv 0,1 \pmod{3}$. (Recall that if $x \equiv y \pmod{a}$, then $x^k \equiv y^k \p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/218915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
How did my teacher gets this result? Algebra/Limits $$\begin{align*}&\lim_{x\to0}\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt[3]{1+x}-\sqrt[3]{1-x}}=\\
&\lim_{x\to0}\left(\frac{(1+x)-(1-x)}{(1+x)-(1-x)}\cdot\frac{\sqrt[3]{(1+x)^2}+\sqrt[3]{(1+x)(1-x)}+\sqrt[3]{(1-x)^2}}{\sqrt{1+x}+\sqrt{1-x}}\right)
\end{align*}$$
My teacher was... | To get from the first line to the second line, your teacher is multiplying by
$$
\frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}} \cdot \frac{\sqrt[3]{(1+x)^2} + \sqrt[3]{(1+x)(1-x)} + \sqrt[3]{(1-x)^2}}{\sqrt[3]{(1+x)^2} + \sqrt[3]{(1+x)(1-x)} + \sqrt[3]{(1-x)^2}}.
$$
Why did he make this choice? He is using t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/219052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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A couple of GRE questions Look for help with the following GRE questions
Question 1. If $C$ is the circle in the complex plane whose equation $|z|=\pi$, oriented counterclockwise, find the value of the integral $\oint_C(\cos z-z\cos\frac{1}{z})dz$.
Question 2. How to show the sequence $\{x_n\}_{n=1}^\infty$ definted by... | For the first, $\oint_C \cos z dz = 0$ since $\cos$ is an entire function. Thus, only the term $z\cos\frac{1}{z}$ is relevant. That term has a single pole at $z=0$, and by the residue theorem you therefore have $$
\oint_C z\cos\frac{1}{z} dz = i2\pi R
$$
where $R$ is the residue (i.e. the coefficient of $z^{-1}$ in t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/219769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
What are the steps to simplify the following modulus expression? I have no clue how to do this exactly. Is there a systematic way of doing this or you just have to do it by trial and error?
$n^2 \equiv 9 \pmod {72}$
to $n \equiv a \pmod b$?
| Note that $n^2\equiv 9 \pmod {72}$ is equivalent to
$$
(n-3)(n+3)\equiv 0 \pmod {72}
$$
So it is sufficient to solve the simultaneous congruences
$$
(n-3)(n+3)=0\pmod 8
$$
and
$$
(n-3)(n+3)\equiv 0 \pmod 9
$$
The second congruence is true iff $n$ is divisible by $3$, because then both $n-3$ and $n+3$ are divisible by $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 0
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Determine the degree of a field extension I have to determine the degree of $\mathbb{Q}\left(\sqrt{2},\sqrt{3}\right)$
over $\mathbb{Q}$ and show that $\sqrt{2}+\sqrt{3}$ is a primitive
element ? Could someone please give me any hints on how to do that
?
| Clearly $[\mathbb Q(\sqrt 2):\mathbb Q]\le 2$ becasue of the polynomial $X^2-2$ and $[\mathbb Q(\sqrt 2,\sqrt 3):\mathbb Q(\sqrt 2)]\le 2$ because of the poylnomial $X^2-3$.
In fact, $\sqrt 2\notin \mathbb Q$ implies $[\mathbb Q(\sqrt 2):\mathbb Q]=2$.
We also have $\sqrt 3\notin \mathbb Q(\sqrt 2)$ because $(a+b\sqrt ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/225418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Prove that $\lim\limits_{x\to 0}\frac{x^2-1+\cos^2x}{x^4+x^3\sin x}=\frac{1}{6}$ without De L'Hôpital Rule or Taylor Expansion? Any ideas how to show $\lim\limits_{x\to 0}\dfrac{x^2-1+\cos^2x}{x^4+x^3\sin x}=\lim\limits_{x\to 0}\dfrac{x-\sin(x)}{x^3}=\dfrac{1}{6}$ without using the De L'Hôpital rule (or proving a speci... | Squeeze Theorem.
First, there is some factoring and cancellation:
$$
\begin{align}
\frac{x^2-1+\cos^2x}{x^4+x^3\sin x} &=\frac{x^2-(1-\cos^2{x})}{x^3(x+\sin x)}\\
&=\frac{x^2-\sin^2(x)}{x^3(x+\sin x)}\\
&=\frac{(x-\sin x )(x+\sin x)}{x^3(x+\sin x)}\\
&=\frac{x-\sin x}{x^3}\\
\end{align}
$$
The function $f$ with $f(x)=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/226867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
General term of this interesting sequence A sequence of positive integer is defined as follows
*
*The first term is $1$.
*The next two terms are the next two even numbers $2$, $4$.
*The next three terms are the next three odd numbers $5$, $7$, $9$.
*The next $n$ terms are the next $n$ even numbers if $n... | The general term is
$$\tag1a_n=2n-\left\lceil \sqrt{2 n}-\frac12\right\rceil.$$
Why?
We have $a_{n+1}=a_{n}+2$ unless there is an integer $\ge\frac12 +\sqrt{2n}$ and $<\frac12+\sqrt{2(n+1)}$.
We need $a_{n+1}=a_n+1$ iff $n$ is one of the numbers $1, 3, 6, 10, \ldots$, i.e. a number of the form $k \choose 2$. This is ac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/229161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Prove $\frac 1{x+1}<\ln\left(\frac{x+1}x\right)<0.5\left(\frac 1x+\frac 1{x+1}\right)$ I looking for help with proving the following inequality. Any relevant logarithmic identities would be great. Tried differentiating and taking limits and I'm lost as to how to approach this.
$$\frac 1{x+1}<\ln\left(\frac{x+1}x\right... | For LHS:use this well konw inequality
$$\dfrac{x}{x+1}\le\ln{(1+x)},x>-1\Longrightarrow \ln{\left(\dfrac{x+1}{x}\right)}=\ln{\left(1+\dfrac{1}{x}\right)}\ge\dfrac{\dfrac{1}{x}}{1+\dfrac{1}{x}}=\dfrac{1}{x+1}$$
For RHS: we only prove
$$2\ln{(1+x)}<x+\dfrac{x}{x+1}=\dfrac{x^2+2x}{x+1}$$
let
$$g(x)=2\ln{(1+x)}-\dfrac{x^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/229599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Power Series: Ratio test, but $x$ cancels out?!? The question is:
Determine the interval of convergence of the power series
$$\sum_{n=1}^\infty\left(\frac{2n+1}{n^2+1}\right)(2x+1)^{12}$$
My attempt at an answer:
$$u_n=\frac{(2n+1)}{(n^2+1)}(2x+1)^{12}$$
Applying the ratio test:
$$\begin{align}
\require{enclose}
\frac{... | If the series really is $$\sum_{n=1}^\infty\left(\frac{2n+1}{n^2+1}\right)(2x+1)^{12}\;,$$ the loss of $x$ is obviously right: this is simply
$$(2x+1)^{12}\sum_{n=1}^\infty\frac{2n+1}{n^2+1}\;,$$
which is either $a(2x+1)^{12}$ for $$a=\sum_{n=1}^\infty\frac{2n+1}{n^2+1}$$ if that series converges, or undefined if it do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/229942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove $\lim_{n \to +\infty} \sqrt{n}\int_0^\pi{\cos(\frac{t}{2})^n}dt>0$ I want to prove
$$\lim_{n \to +\infty}\sqrt{n}\int_0^\pi{\cos\left(\frac{t}{2}\right)^n}dt>0.$$
First, I consider $$\lim_{n \to +\infty}\sqrt{n}\int_0^\pi{\cos\left(\frac{t}{2}\right)^n}\sin\left(\frac{t}{2}\right)dt,$$ which is smaller th... | By a change of variables $t\mapsto 2x$, you want to show $$\mathop {\lim }\limits_{n \to + \infty } 2\sqrt n \int_0^{\frac{\pi }{2}} {{{\cos }^n}x} dx > 0.$$
Now, you need to evaluate $$I\left( n \right) = \int_0^{\frac{\pi }{2}} {{{\cos }^n}x} dx$$
For $n\geq 2$ we can reduce this integral with integration by parts
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/231526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Encode the message $[1,1,0,1,1,0,1]$ in BCH code based on the field $\mathbb F = \frac{\mathbb Z_{2}[x]}{x^4+x+1}$ So here's what I understand so far:
$\mathbb F = \frac{\mathbb Z_{2}[x]}{x^4+x+1} = GF(16)$
The code is written as $[x^{14},x^{13},x^{12},x^{11},x^{10},x^{9},x^{8}$ $|$ $x^{7},x^{6},x^{5},x^{4},x^{3},x^{2}... | Yes, the question is somewhat incomplete (as pointed by Dilip Sarwate and Jyrki Lahtonen).
This is a 2-error correcting (primitive narrow-sense) BCH code. The generator polynomial is the smallest binary polynomial that has $\alpha, \alpha^2, \alpha^3, \alpha^4$ as its roots. This is precisely the LCM of the minimal po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/234617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Fibonacci conjecture: $(F_{n+5})^2 - (F_n)^2 = 3((F_{n+3})^2 - (F_{n+2})^2) + 8 F_{n+2} F_{n+3} $. So this is the question I have
The Fibonacci sequence is a recurrence system given by $$F_1 = 1, \ F_2 = 1, \ F_{n+2} = F_{n+1} + F_n \qquad (n = 1, 2, 3, \ldots).$$ This question concerns the following conjecture: $$(F_... | He substituted $F_{n+3}+F_{n+2}$ for $F_{n+4}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inequality with constraint I've been trying to prove the following inequality without success.
For $a,b,c \in \mathbb{R}$ such that $abc=1$, prove that:
$$\frac{1}{a^2+a+1}+\frac{1}{b^2+b+1} + \frac{1}{c^2+c+1} \geq 1$$
| Multiply both sides of the inequality by $(a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1)$ to get
$$
\begin{align*}
&(b^2 + b + 1)(c^2 + c + 1) + (a^2 + a + 1)(c^2 + c + 1) + (a^2 + a + 1)(b^2 + b + 1)\\
\geq &(a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1).
\end{align*}
$$
Suffer through the algebra (or enlist Maple's help) to simpl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Visualization of Singular Value decomposition of a Symmetric Matrix The Singular Value Decomposition of a matrix A satisfies
$\mathbf A = \mathbf U \mathbf \Sigma \mathbf V^\top$
The visualization of it would look like
But when $\mathbf A$ is symmetric we can do:
$\begin{align*}
\mathbf A\mathbf A^\top&=(\mathbf U\ma... | Singular value decomposition
Start with a matrix with $m$ rows, $n$ columns, and rank $\rho$,
$$
\mathbf{A}\in\mathbb{C}^{m\times n}_{\rho}
$$
which has the singular value decomposition
$$
\mathbf{A} =
\mathbf{U} \, \Sigma \, \mathbf{V}^{*} =
%
\left[ \begin{array}{cc}
\color{blue}{\mathbf{U}_{\mathcal{R}\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/243811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Best way to discuss a system of linear equations that depends on two parameters. I know there is no specific way, but what is, in your opinion, the best way to discuss a system of linear equations like:
$$
\left\{
\begin{array}{l}
ax+by+z = 1\\
x+aby+z=b\\
x+by+bz=1
\end{array}
\right.
$$
$$(A|B) = \left(\begin{array}... | If $b=0$ the system leads to also $a=2$ and then the solution is $x=1,z=1$ with $y$ arbitrary.
If $a=1$ it leads to also $b=1$ and in this case the system becomes the single relation $x+y+z=1$.
Now if we assume that $b$ is not zero (since the case $b=0$ already dealt with), but that $-2+b+ab=0$, then we have $a=(2-b)/b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What are $2222^{5555}+5555^{2222} \pmod 7$ and $9^{2n+1}+8^{n+2} \pmod{73}$? Tell me hint for solve :
1) $ 2222^{5555}+5555^{2222} \equiv \mathord? \pmod 7$
2) $ 9^{2n+1}+8^{n+2} \equiv \mathord ?\pmod{73}$
thank you.
| For problem 1) You can show by elementary division that $$2222 \equiv 3 \mod 7$$
Now $$3^6 \equiv 1 \mod 7$$
And therefore (all mod 7)
$$2222^{5555} \equiv 3^{5555} \equiv 3^{6\times 925+5} \equiv 3^5 \mod 7 $$
Now since inverse of 3 is 5 in mod 7, the above becomes $3^6.3^{-1}\equiv 5 \mod 7$
Similarly you can show $5... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to calcualate how many unique set of 6 can i have in a given set. Hello my question is quite simple i would think but i just cant seem to find an answer. I have a set of $\{1,2,3,4,5,6,7,8,9,10\}$ and i would like to calculate how many unique given sets of $6$ can i get from this set.
In other words for the number... | Binomial coefficients count the number of distinct subsets of $k$ elements from a set containing $n$ elements. The notation for this is $\binom{n}{k}$ which is equal to $\frac{n!}{k!(n-k)!}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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How come $2 \times 3^k + 3^k = 3 \times 3^k$ I get something else
$$\begin{align*}
&2 \times 3^k + 3^k=\\
&2 \times 3^k \times 2=\\
&4 \times 3^k
\end{align*}$$
What does $3^k + 3^k$ give exactly?
| $2\times 3^k+3^k=(2+1)\times3^k=3\times3^k$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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How to integrate $(x^{1/2} + 5 x^{1/3})^{-1}$? $$ \int_0^1 \frac{dx}{x^\frac12+5x^\frac13} $$
Let $$u = 5+x^\frac16 $$
I have change x into u, and find $$\int_5^6 \frac{7}{6u(u-5)^3} $$
Then, i don't know how to integrate it.
| Putting $x=(t-5)^6, dx=6(t-5)^5$ and $x=0\implies t=5,x=1\implies t=6$
We get $$ \int_0^1 \frac{dx}{x^\frac12+5x^\frac13}=\int_5^6 \frac{6(t-5)^3dt}{t} =6\int_5^6 (t^2-15t+75-125\frac1t)dt=6(\frac{t^3}3-15\frac{t^2}2+75t-125\log t)\mid_5^6$$
$$=2(6^3-5^3)-45(6^2-5^2)+450(6-5)-750\log\frac65$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve${{\int_{-1}^{\infty }{\left( \frac{{{x}^{4}}}{1+{{x}^{6}}} \right)}}^{2}}dx$ Please help me to evaluate the integral:
$\displaystyle {{\int_{-1}^{\infty }{\left( \frac{{{x}^{4}}}{1+{{x}^{6}}} \right)}}^{2}}dx$
Thanks.
| Notice that:
$$x^8 = (x^6 + 1)x^2 - x^2$$
So that your integrand takes the form:
$$\int\frac{x^2}{1+x^6}dx - \int\frac{x^2}{(1+x^6)^2}dx$$
Now substitute $u = x^3, du = 3 x^2 dx$:
$$\int\frac{1}{3(1+u^2)}du - \int\frac{1}{3(1+u^2)^2}du$$
The first integral is a multiple of $\arctan(u)$, and the second can be solved div... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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No natural numbers sucht that $x+3y+5z=2012$ and $x^2+y^2+3z^2=2013$. Prove that there are not exist the natural numbers, $x,y,z$ such that $x+3y+5z=2012$ and $x^2+y^2+3z^2=2013$.
tell me please if my proof is ok ?
$x^2+y^2=3\cdot671-3z^2=3(671-z^2)=3k$ but if we want we can find out $z$. $671 \geq z^2$ so $z=\overl... | You are fine to $x^2+y^2=3(671-z^2)=3k$, but I don't know what you mean by $z=\overline{0\ldots25}$. You are correct that perfect squares are $4k$ or $4k+1$ (but be careful about reusing $k$. You might confuse yourself thinking they are the same). This means $x^2+y^2 \equiv 0,1,2 \pmod 4$ but that doesn't say anythi... | {
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"timestamp": "2023-03-29T00:00:00",
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Second order partial differential equation Second order partial differential equation:
$$(x-y)^2/4 u_{xx} + (x-y) \sin(x^2+y^2) u_{xy} + \cos(x^2+y^2) u_{yy} +\dots=0$$ is
a. Elliptic in $\{(x,y): x≠y, x^2+y^2<\frac{\pi}{6}\}$
b. Hyperbolic in $\{(x,y): x≠y, \frac{\pi}{4}<x^2+y^2<\frac{3\pi}{4}\}$
c. Elliptic in ... | compute the expression $S^2-4RT$ . here $S=(x-y)\sin(x^2+y^2)$ , $R=(x-y)^2/4$ , $T=\cos^2(x^2+y^2)$. after calculating we have
$S^2-4RT=(x-y)^2\cos 2(x^2+y^2)$
so the option ($3$) is correct
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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how to compute the matrix polynomial Let $A$ be the matrix $$A = \begin{pmatrix} 1 & \sqrt{2}\\ -\sqrt{2} & -1\\ \end{pmatrix}$$
Compute the matrix $B = 3A -2A^2 - A^3 -5A^4 + A^6$.
Could any one give me any hint for this one? I have calculated the eigenvalues they are $(1+\sqrt{2}i),(1-\sqrt{2}i)$
| We have the characteristic polynomial $ch_A(x)=\begin{vmatrix}1-x&\sqrt{2}\\-\sqrt{2}&1-x\end{vmatrix}=(x-1)^2+2=x^2-2x+3$.
Then by Cayley-Hamilton theorem we know that $A^2-2A+3=0$.
So if you divide your polynomial $p(x)$ by $(x^2-2x+3)$ and you get $p(x)=q(x)(x^2-2x+3)+r(x)$, you only need to calculate $r(A)$, i.e., ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Do there exist any odd prime powers that can be represented as $n^4+4^n$? Well, I wrote up a solution on it, but according to the place I found the problem, it isn't quite correct. Ah, I'm simply hoping someone will point out where I got wrong.
Now, let, $n^4+4^n = p^k$, where $p$ is an odd prime and $k$ is a positive ... | There is the small counterexample
$$5^1=1^4+4^1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/261925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
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incircle and circumcircle Given a triangle $\Delta ABC$ with circumcenter $O$, and incenter $I$.
If $∠ABC = 45°$, $OI = d$, and $c − b = d√2$,
Find the value of $\sin A$.
I've thought of using the fact that $d^2 = R(R-2r)$ where $R$ is the circumradius and $r$ is the inradius. But I didn't know where to apply it.
| Seat your belt, it will be great...
You can use your $d^2 = R(R-2r)$, which gives you
$(c-b)^2 = 2d^2 = 2R(R-2r) = 2R^2-4rR$
Then, you remark that the area of the triangle can be obtained with $r$ or with $R$ :
$\displaystyle\text{Area} = \frac{(a+b+c)r}{2}$ and $\displaystyle\text{Area} = \frac{ab\sin{C}}{2} = \frac{a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing that: $(\frac{a}{b+c})^2+(\frac{b}{a+c})^2+(\frac{c}{a+b})^2+\frac{10abc}{(a+b)(b+c)(c+a)}\ge 2$ Let a;b;c>0. Prove:
$$\left(\frac{a}{b+c}\right)^2+\left(\frac{b}{a+c}\right)^2+\left(\frac{c}{a+b}\right)^2+\frac{10abc}{(a+b)(b+c)(c+a)}\geq 2$$
I think
$$\frac{2a}{b+c}=x;\frac{2b}{c+a}=y;\frac{2c}{a+b}=z$$
We ha... | I collected a solution:
Need to prove: $x^2+y^2+z^2+5xyz \ge 8$
Put: $x+y+z=p;xy+yz+zx=q;xyz=r$
We have: $q+r=4$
Need to prove inequality is equivalent to:
$p^2-2q+5r \ge 8 \Leftrightarrow p^2-7q+12 \ge 0$
*)If: $p \le 4$. Applying Schur's inequality; we have:
$r \ge \dfrac{p(4q-p^2)}{9} \Rightarrow 4 \ge q+\dfrac{p(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Range of a function $f(x)$
As x varies over all real numbers, the range of the function $$f(x)=\frac{x^2-3x+4}{x^2+3x+4}$$ is (1) $[\frac{1}{7},7]$, (2) $[-\frac{1}{7},7]$, (3) $[-7,7]$ (4) $(-\infty,\frac{1}{7})\bigcup(7,\infty)$.
Trial:$$\begin{align} \frac{x^2-3x+4}{x^2+3x+4} &=\frac{(x-\frac{3}{2})^2+\frac{7}{4}... | Let $\dfrac{x^2-3x+1}{x^2+3x+1} = y$. We then have $$(y-1)x^2 + 3(y+1)x + (y-1) = 0$$
For this to have a solution, we need discriminant of the quadratic to be non-negative. Hence, we get
$$9(y+1)^2 - 4 (y-1)^2 \geq 0 \implies 5y^2 +26y+5 \geq 0 \implies (5y+1)(y+5) \geq 0$$
This gives us $y \geq -\dfrac15$ and $y \leq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the limit of $(x_n)$ defined by $x_{n+1}=c_nu(x_n)$ Find $\displaystyle \lim_{n\rightarrow \infty }x_n$ :
$$\left\{\begin{matrix}x_1=a>0\\ \\ x_{n+1}=\frac{2x_n\cdot \cos\left(\frac{\pi}{2^n+1}\right)}{x_n+1}\end{matrix}\right.$$
I have tried that :
Let $a_n=\dfrac{1}{x_n}$ . So :
$$a_{n+1}=\frac{1}{2\cos\left(\fr... |
$$\lim\limits_{n\to\infty}x_n=1$$
To show this, note that: first, $x_n\gt0$ for every $n\geqslant1$; second, $x_n\leqslant x^0_n$ for every $n\geqslant1$, where $x^0_1=a$ and
$$
x^0_{n+1}=\frac{2x^0_n}{x^0_n+1},
$$
for every $n\geqslant1$; third, $x^0_n\to1$ when $n\to\infty$. Thus, $\limsup\limits_{n\to\infty}x_n\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$\lim_{x \to 0}\frac{|x|\sin \left(\frac{1}{3 \sqrt{x}}\right)}{\sqrt{x^4+4x^2+7}}$
Find $$\lim_{x \to 0}\frac{|x|\sin \left(\frac{1}{3 \sqrt{x}}\right)}{\sqrt{x^4+4x^2+7}}$$
I know that $\lim_{x \to 0} \frac{\sin x}{x}=1$ But here $\sin \left(\frac{1}{3 \sqrt{x}}\right)$ is given when $x \to 0$. Need help.
| The limit is zero, because $|x| \sin{\frac{1}{3 \sqrt{x}}} \rightarrow 0$ as $x \rightarrow 0$. (The denominator is nonzero in this limit.).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/269143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Direct proof that $(5/p)=1$ if $p\equiv 1\pmod{5}$. I'm trying to show without use of quadratic reciprocity that $(5/p)=1$ if $p\equiv 1\pmod{5}$. If $p\equiv 1\pmod{5}$, then there exists some $x\in U(\mathbb{Z}/p\mathbb{Z})$ with order $5$.
I note that modulo $p$, $(x+x^4)^2+(x+x^4)-1=0$, since
$$
(x+x^4)^2+(x+x^4)-... | Multiply your relation by $4$. We get
$$4(x+x^4)^2+4(x+x^4)-4\equiv 0\pmod{p}.\tag{$1$}$$
Complete the square: $(1)$ is equivalent to $\left(2(x+x^4)+1\right)^2\equiv 5\pmod{p}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/271345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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$\epsilon$-$\delta$ proof that $f(x) = x \sin(1/x)$, $x \ne 0$, is continuous I'm doing an exercise that asks me to prove that $f$ is continuous using a $\epsilon$-$\delta$ proof. I have that
$$
f(x) = \begin{cases}
x\cdot \sin \frac1x,&x\neq 0
\\
0,&x = 0
\end{cases}
$$
I've already managed to show this property for... | Although the answer given by @Nameless is complete in itself, a slight change in the choice of added terms in the triangle inequality helps a lot. A modified version of his answer is as follows.
Consider
\begin{align*}
\big|f(x)-f(a)\big|&=\left|x\sin\frac{1}{x}-a\sin\frac{1}{a}\right|\\
&=\left|x\sin\frac{1}{x}-x\sin\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/274594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$ Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$
where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums... | Interestingly, $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}H_{n}^{-}}{n} = \frac{\zeta(2)}{2} {\color{red}{+}} \frac{\log^{2} (2)}{2}$$ where $H_{n}^{-}$ are the alternating harmonic numbers defined as $$H_{n}^{-} = \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k} .$$
One way to show this is to notice that $$ \begin{align} \log (2) - H_... | {
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"url": "https://math.stackexchange.com/questions/275643",
"timestamp": "2023-03-29T00:00:00",
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Decomposition of polynomial into irreducible polynomials I'm preparing to my algebra exam. And I have problem and I have no idea how to solve it.
Given polynomial
$$x^4+4x^3+4x^2+1.$$
The task is find expansion of the polynomial as a product of irreducible polynomials in $\mathbb{R}$.
I will be happy if you show ... | It suffices to show that $f(x)=x^4+4x^3+4x^2+1$ has no linear factors over $\mathbb{Z}_3$: $f(0)=f(1)=1$ and $f(2)=2$, so $f(x)$ has no linear factors. Then $f(x)$ must factor to two quadratic polynomials: $$f(x)=(ax^2+bx+c)(ux^2+vx+w)$$ We then have that $au=1$. Multiplying the first polynomial by $u$ and the second... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "7",
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Calculate $\int\frac{1}{(x+1)\sqrt{1+x^2}} dx $ How can I calculate the following integral :
$$\int\frac{1}{(x+1)\sqrt{1+x^2}} dx $$
I try to write the integral like :
$$\int\frac{1+x-x}{(x+1)\sqrt{1+x^2}}=\int\frac{1}{\sqrt{1+x^2}}-\int\frac{x}{(x+1)\sqrt{1+x^2}}=\int\frac{1}{\sqrt{1+x^2}}-\int\frac{(\sqrt{x^2+1})'}{... | Substitute $x=\frac{1-t}{1+t}$
$$\int\frac{1}{(x+1)\sqrt{1+x^2}} dx =-\int\frac1{\sqrt{2(1+t^2)}}dt=-\frac1{\sqrt2}\sinh^{-1}t
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $a, b$ and $c$ be the lengths of the sides of an arbitrary triangle. Pick out the true statements. Let $a, b$ and $c$ be the lengths of the sides of an arbitrary triangle. Define
$$x =\frac{ab + bc + ca}{a^2 + b^2 + c^2}.$$
Pick out the true statements.
(a) $1/2 ≤ x ≤ 2$.
(b) $1/2 ≤ x ≤ 1$.
(c) $1/2 < x ≤ 1$. ... | Triangle inequality $$\Rightarrow\left | a-b \right |<c \Rightarrow \sum_{a,b,c}(a-b)^{2}<\sum_{a,b,c}c^{2}\Rightarrow 2\sum_{a,b,c}a^{2}-2\sum_{a,b,c}ab<\sum_{a,b,c}a^{2}\Rightarrow \frac{1}{2}<\frac{\sum\limits_{a,b,c}ab}{\sum\limits_{a,b,c}a^{2}}=x$$ $x\in \mathbb{R}\Rightarrow x^{2}\geq 0$
$$\Rightarrow(a-b)^2\ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Integral of $\frac{x^3-2}{x^3-x}$ As the title says, I need some help with this function:
$$ \int\frac{x^3-2}{x^3-x}$$
I tried it with different versions/forms of the function to get it with substitution or partiel integration:
$$ \int\frac{x^3-2}{x^3-x} = \frac{x^3-2}{x(x-1)(x+1)} = \frac{x^3}{x^3-x} - \frac{2}{x^3-x}... | try this
$$\frac{x^3-2}{x^3-x} = 1 + \frac{x-2}{x^3-x}$$
Now use partial fraction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/280724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find $x,y,z$ if $3^{x^4+y^2}+3^{y^4+z^2}+3^{z^4+x^2}=3^7$. How can I solve the following system with $x,y,z$ real numbers:
$$x^2+y^2+z^2=6$$
$$3^{x^4+y^2}+3^{y^4+z^2}+3^{z^4+x^2}=3^7.$$
I observe that $x=y=z=\sqrt{2}$ and I feel it must apply the inequality $AM \geq GM$ but I don't know how to do.
Thanks :)
| Applying $AM\geq GM$ to your second equation you get
$$3^6\geq ({3^{x^4+y^2+y^4+z^2+z^4+x^2}})^{1/3}=(3^{x^4+y^4+z^4})^{1/3}\cdot(3^{x^2+y^2+z^2})^{1/3}=(3^{x^4+y^4+z^4})^{1/3}\cdot 3^2$$
and therefore $$3^4\geq 3^{\frac{x^4+y^4+z^4}{3}}$$
Now we know that $\sqrt{\frac{x^4+y^4+z^4}{3}}\geq \frac{x^2+y^2+z^2}{3}=2$ wher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Proving an inequality for positive numbers $a, b, c$
Let be $a,b,c$ positive numbers such that $a+b+c=3$. Prove that
$$\frac{b+c+bc}{a^2+b^3+c^4}+\frac{c+a+ca}{b^2+c^3+a^4}+\frac{a+b+ab}{c^2+a^3+b^4} \le 3$$
| By Holder
$$\sum_{cyc}\frac{a+b+ab}{c^2+a^3+b^4}=\sum_{cyc}\frac{9(c^2+a+1)(a+b+ab)}{(c^2+a+1)(c^2+a^3+b^4)(1+1+1)(1+1+1)}\leq$$
$$\leq\frac{9\sum\limits_{cyc}(c^2+a+1)(a+b+ab)}{(a+b+c)^4}=\frac{1}{9}\sum_{cyc}(a^2b+a^2c+a^2bc+a^2+ab+a^2b+2+ab)=$$
$$=\frac{1}{9}\sum_{cyc}(2a^2b+a^2c+a^2bc+a^2+2ab+2)=\frac{1}{9}\sum_{cy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
integral via partial fractions I have to find the integral of this rational function:
$$\frac{x^5}{(x-1)^2(x^2-1)}$$
Since the power in the numerator is bigger than the power in the denominator, I have to divide the first one by the second one. So I divide it and I have $(x+2)$ and the quotient $(4x^3-2x^2-3x-2)$. Now ... | Assuming you have done your algebra correctly, you have arrived at $${4x^3-2x^2-3x-2\over(x-1)^3(x+1)}={A\over x-1}+{B\over(x-1)^2}+{C\over(x-1)^3}+{D\over x+1}$$ and you want to find $A,B,C,D$. Clear fractions to get $$4x^3-2x^2-3x-2=A(x-1)^2(x+1)+B(x-1)(x+1)+C(x+1)+D(x-1)^3$$ Now you have some options. You can multip... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Inequality: $(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27$
Let be $a,b,c \geq 0$ such that: $a^2+b^2+c^2=3$.
Prove that:
$$(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27.$$
I try to apply $GM \leq AM$ for $x=a^3+a+1$, $y=b^3+b+1,z=c^3+c+1$ and
$$\displaystyle \sqrt[3]{xyz} \leq \frac{x+y+z}{3}$$ but still nothing.
Thanks :-)
| This will have derivatives. Substitute $a=\sqrt{x}, b=\sqrt{y}$ and $c=\sqrt{z}$. Then $x+y+z=3$. Consider the function $f(x)=\ln(x\sqrt{x}+\sqrt{x}+1)$. It is concave.
Hence Jensen yields: $f(x)+f(y)+f(z)<=3f(1)=3\ln{3}$
This is equivalent to what is asked.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/283895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
"answer_count": 8,
"answer_id": 2
} |
Inequality on the side lengths of a triangle: $\left| \frac{a}{b} + \frac{b}{c} + \frac{c}{a} - \frac{a}{c} - \frac{b}{a} - \frac{c}{b} \right| < 1$. This problem is taken from the Kosovo Mathematical Olympiad for Grade-$ 10 $ students.
Let $ a $, $ b $ and $ c $ be the lengths of the edges of a given triangle.
How can... | You can write the question as
$|\frac{a^2c+b^2a+c^2b-b^2c-c^2a-a^2b}{abc}|<1$. This means you would like to show that
$-1<\frac{a^2c+b^2a+c^2b-b^2c-c^2a-a^2b}{abc}<1$. Since $a,b,c$ are the sides of a triangle, you have that $a,b,c>0$. This means you want to show $-abc<a^2c+b^2a+c^2b-b^2c-c^2a-a^2b<abc$. We write th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 0
} |
Determinant of a block matrix with $\mathrm{Id}$ and $0$ in the diagonal How to compute the determinant $\det A$ depending on $B$ and $C$, where
$$ A = \left(\begin{matrix}\mathrm{Id} & B \\ C & 0 \end{matrix} \right), $$
a) when $C$ is square,
b) $C$ has more rows than columns
| Hints:
*
*note that $$\begin{pmatrix} \text{Id} & B \\
C & 0 \end{pmatrix} = \begin{pmatrix} \text{Id} & 0 \\
C & \text{Id} \end{pmatrix} \cdot\begin{pmatrix} \text{Id} & B \\
0 & - C\cdot B\end{pmatrix}.$$
*furthermore, in general $$\det\begin{pmatrix}A& 0\\ C& D\end{pmatrix} = \det\begin{pmatrix}A& B\\ 0& D\end{p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/286074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to prove $a_{n} < 2$ if $\displaystyle a_{n+2}=a_{n+1}+\frac{a_{n}}{3^n}$ Let $(a_{n})_{n \geq1}$ be a real sequence such that $a_{1}=a_{2}=1$ and $\displaystyle a_{n+2}=a_{n+1}+\frac{a_{n}}{3^n}, n\geq 1$.
Prove that $a_{n} < 2, \forall n \geq 1.$
I write $$\sum a_{k+2}-a_{k+1}=\sum \frac{a_{k}}{3}$$
and I obt... | Claim:
$$a_n < 2-\frac{1}{3^n} \,.$$
$P(1), P(2)$ are easy to check.
Inductive step:
$$a_{n+1}= a_{n+1}+\frac{a_n}{3^n} \leq 2-\frac{1}{3^{n+1}}- \frac{2-\frac{1}{3^n}}{3^n}=2-\frac{1}{3^{n+1}}- \frac{2}{3^n}+\frac{1}{9^n} $$
If we can prove taht
$$2-\frac{1}{3^{n+1}}- \frac{2}{3^n}+\frac{1}{9^n} < 2-\frac{1}{3^{n+2}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/287563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Prove this product: $\prod\limits_{k=2}^ n {\frac{k^2+k+1}{k^2-k+1}}=\frac{n^2+n+1}{3}$ How to prove this product?
$$\prod\limits_{k=2}^ n {\frac{k^2+k+1}{k^2-k+1}}=\frac{n^2+n+1}{3}$$
| HINT: It never hurts to gather some data by doing some actual computation:
$$\begin{array}{c|l}
n&\prod_{k=2}^n\frac{k^2+k+1}{k^2-k+1}\\ \hline
2&\frac73\\
3&\frac{\color{red}7}3\cdot\frac{13}{\color{red}7}\\
4&\frac{\color{red}7}3\cdot\frac{\color{blue}{13}}{\color{red}7}\cdot\frac{21}{\color{blue}{13}}\\
5&\frac{\col... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/287880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 2
} |
Prove $|\frac{1}{n} - \frac{1}{x}| \le \frac{1}{n^2}$ Prove $|\frac{1}{n} - \frac{1}{x}| \le \frac{1}{n^2}$ for all $n \le x \le n+1$, using the mean value theorem applied to $f(x) = \frac{1}{x}$
Immediately, I can recognize some components of the mean value theorem. $\frac{1}{n^2}$ likely comes from the slope of $f$ ... | If $x = n$, then the left hand side is $0$ and the result holds. So let's assume that $n < x \leq n+1$. Then applying the mean-value theorem to the function $f(t) \colon= 1/t$ for $t \in [n, x]$, we find that there is a real number $c \in (n,x)$ such that
$$ f(x)-f(n) = f\prime(c) \cdot (x-n) $$
or
$$\frac{1}{x}-\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/288580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Find $\lim_{x \rightarrow 0} \frac{e^{\sin x} - e^x}{\sin^3 2x}$ I have to find $\displaystyle\lim_{x \rightarrow 0} \frac{e^{\sin x} - e^x}{\sin^3 2x}$ using Taylor polynomials.
Here's what I've done so far:
*
*$e^x = 1 + x + \frac{1}{2} x^2 + \frac{1}{6} x^3 + o(x^3)$
*$\sin x = x - \frac{1}{6} x^3 + o(x^4)$
*$e... | $\mathbb{e}^x = 1 + x + \frac{1}{2} x^2 + \frac{1}{6} x^3 + o(x^3)$ and $\sin x=x-\frac{1}{6}x^3+o(x^3)$
$\Rightarrow \mathbb{e}^{\sin x}=1+x-\frac{1}{6}x^3+\frac{1}{2}x^2+\frac{1}{6}x^3+o(x^3)=1+x+\frac{1}{2}x^2+o(x^3)$
$\Rightarrow \mathbb{e}^{\sin x}-\mathbb{e}^x=-\frac{1}{6}x^3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/288893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Minimum of $\left| \sin x- 1\right| + \left|\sin x- 2\right| + \left| \sin x -3\right| + \left| \sin x+1\right|$ I would like to know the minimum value of $$\left| \sin x- 1\right| + \left|\sin x- 2\right| + \left| \sin x -3\right| + \left| \sin x+1\right|$$
for $x \in \mathbb{R}$.
| For every $x \in \mathbb{R}$ we have
$$
-1 \le \sin x\le 1 \quad \forall\ x \in \mathbb{R},
$$
and therefore:
\begin{eqnarray}
f(x)&:=&|\sin x-1|+|\sin x-2|+|\sin x-3|+|\sin x+1|\\
&=&(1-\sin x)+(2-\sin x)+(3-\sin x)+(\sin x+1)\\
&=&5+2(1-\sin x).
\end{eqnarray}
The minimum is therefore equal to $5=f(\pi/2)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/289185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 3
} |
$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$ Evaluate Integral Anyone remember the method for this? I think this should been done on the site
$$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$$
| I think @kiwi ment
$$
\fbox {$I$} = \int_0^\infty \frac {\ln x}{x^2+a^2} dx = \left | u = \frac {a^2}x \Longrightarrow\left\{\begin{array}{c}
\ln x = 2 \ln a - \ln u \\
dx = -\frac {a^2du}{u^2}
\end{array}\right\} \right | = -\int_\infty^0 \frac{2\ln a - \ln u}{\frac {a^4}{u^2}+a^2}\frac {a^2 du}{u^2} = \\
2\int_0^\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/290200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 5,
"answer_id": 3
} |
Help solving summation series of a recursive function Yesterday in class, we were analyzing the Karatsuba multiplication algorithm and how it applies to recurrence equations. Time ran short, and I feel I missed how to solve the final summation.
First, we defined the recurrence equation as
$$T(n) = 3T \left(\frac{n}{2}... | This recurrence has the nice property that we can compute explicit values for $T(n)$ the same way as was done here, for example.
Let $$n = \sum_{k=0}^{\lfloor \log_2 n \rfloor} d_k 2^k$$ be the binary digit representation of $n.$ It is not difficult to see that with $T(0)=0$ we have $$ T(n) = 4 \sum_{j=0}^{\lfloor \lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/291681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Comparing $\large 3^{3^{3^3}}$, googol, googolplex How to show that $\large 3^{3^{3^3}}$ is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$).
Thanks much in advance!!!
| $$3^{3^{3^3}} > 3^{300} > 10^{100}$$ since $$3^{3^3} > 3^7 = 3 \cdot (3^3)^2 > 3 \cdot 10^2 = 300$$ since $$3^3 > 7$$
$$3^{3^{3^3}} < 10^{3^{3^3}} < 10^{10^{100}}$$ since $$3^{3^3} < 3^{100} < 10^{100}$$ since $$3^3 < 100$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/291729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Uniqueness of solution for the functional equation $f(x)=\frac{x}{2}f\left(\frac{x}{2}\right)+h(x)$ Let $h \in \mathcal{C}:=C([-a,a])$, where $a>0$. Prove that there exists a unique function $f \in \mathcal{C}$ such that
$$
f(x)=\frac{x}{2}f\Big(\frac{x}{2}\Big)+h(x)\quad \forall x \in [-a,a].
$$
| If $f$ is a solution then for all $x \in [-a,a]$ we get by applying the functional equation $n$ times
\begin{align*}
f(x) &= \frac{x}{2} f\left(\frac{x}{2}\right) + h(x)\\
&= \frac{x^2}{2^{1+2}} f\left(\frac{x}{4}\right) + \frac{x}{2} h\left(\frac{x}{2}\right) \\
&= \frac{x^2}{2^{1+2+3}} f\left(\frac{x}{8}\right) + \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
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Combinatorics. Please explain me how to do it. In how many different ways three persons A, B, C having 6, 7 and 8 one rupee coins respectively can donate Rs.10 collectively? This isn't a homework question. Please explain me the steps. Thank you!
| Let $a,b,c$ be the amounts that $A,B$ and $C$ give resp. Then we want to count the number of distinct solutions to $a + b + c = 10$ under the condition that $a,b,c$ are positive integers and $a \le 6, b \le 7, c \le 8$.
One way to compute this is to compute the coefficient of $x^{10}$ in the expression $(1+x+\ldots +x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/293610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Trigonometry with multiple angle and exact value of $\tan\pi/5$ By considering the equation $\tan5\theta=0$, show that the exact value of $\tan\pi/5$ is $\sqrt{5-2\sqrt{5}}$.
Do I need to evaluate the multiple angle for $\tan5\theta=0$?
| Using Euler's Formula and Binomial Theorem, we get
$$
\begin{align}
\cos(5\theta)+i\sin(5\theta)
&=\left(\cos(\theta)+i\sin(\theta)\right)^5\\
&=\cos^5(\theta)+5i\cos^4(\theta)\sin(\theta)-10\cos^3(\theta)\sin^2(\theta)\\
&-10i\cos^2(\theta)\sin^3(\theta)+5\cos(\theta)\sin^4(\theta)+i\sin^5(\theta)\tag{1}
\end{align}
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/294884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Induction on binomial Identity: $0\cdot {n\choose 0} + 2\cdot {n\choose 2} + 4\cdot {n\choose4}+\ldots = n\cdot2^{n-2}$ I am having trouble proving the following identity:
$0\cdot {n\choose 0} + 2\cdot {n\choose 2} + 4\cdot {n\choose4}+\ldots = n\cdot2^{n-2}$
Here is what I have so far:
Proof:
Base: Let $n=0$:
LHS: $0\... | Add the first two to get the third:
$$
\begin{align}
\sum_{k=0}^nk\binom{n}{k}&=\sum_{k=0}^nn\binom{n-1}{k-1}&=n(1+1)^{n-1}&=n2^{n-1}\\
\sum_{k=0}^n(-1)^kk\binom{n}{k}&=\sum_{k=0}^n(-1)^kn\binom{n-1}{k-1}&=n(1-1)^{n-1}&=0\\
\sum_{k=0}^{\lfloor n/2\rfloor}2k\binom{n}{2k}&&&=n2^{n-1}\\
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/295019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Determinant of block matrix Here is a determinant of a $(k+m) \times (k+m)$ Block matrix.
\begin{align}
D=\begin{vmatrix}
a_{11} &a_{12} & \cdots & a_{1k} &0 &\cdots &0 \\
a_{21}& a_{22}& \cdots & a_{2k} & 0 &\cdots &0 \\
\vdots& \vdots & & \vdots & \vdots & &\vdots\\
a_{k1} & a_{k2} & \cdots & a_{kk} ... | The discrepancy is that you have to interchange two adjacent rows. From this it is quite clear you have to perform $k\cdot m$ such operations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/297288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Solve for $ x$, $-\frac{1}{2}x^2 + 2x + 5 = 0$ I'm having trouble solving this equation for $x$:
$$-\frac{1}{2}x^2 + 2x + 5 = 0$$
What's the steps to take to solve it?
Thanks.
| $$x^2-4x-10=0$$
$$x^2-2\cdot x\cdot2+2^2-2^2-10=0$$
$$x^2-2\cdot x\cdot2+2^2=4+10$$
$$(x-2)^2=14$$
$$x-2=\pm \sqrt{14}$$
$$x=2\pm \sqrt{14}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/297578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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What are the three cube roots of $1-i$? I am trying to find the three cube roots of $1-i$ and I am unsure of how to proceed. My math teacher recommended I use de Moivre's formula, however I do not know how to set it up. How do you suggest I continue?
de Moivre's formula
$$(\cos x + i\sin x)^n = \cos(nx) + i\sin(nx)$$
| Another way without using DeMoivre's formula that works for cube roots is set
$x=1$ and $y=-i$ and then find any real roots of the following equation
$$\frac{(-64a^9+(48x)a^6+((15(x)^2)-3(3y)^2)a^3+(x)^3)}{-64} = 0$$
This expands to
$$a^9 - \frac{3}{4}a^6 - \frac{21}{32}a^3 - \frac{1}{64} = 0$$
and has three real roots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/298367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
} |
Finding limit of a function ($0$ appears in denominator) This is the function:
$$f(x)= \frac{x^3 + 3x^2+2x}{x^2-2x-8} $$
So I tried to find this limit:
$$\lim_{x\to -2}\frac{x^3 + 3x^2+2x}{x^2-2x-8}=\left[\frac{-8-12-4}{4+4-8}=\frac{-24}{0}\right]= \lim_{x\to -2}\frac{x(x-1)(x-2)}{(x-4)(x+2)}$$
This isn't a case where ... | You have factor wrong your function, just do it in the following way
$$\lim_{x\to -2}\frac{x^3 + 3x^2+2x}{x^2-2x-8} = \lim_{x \to -2} \frac{x(x+1)(x+2)}{(x-4)(x+2)} = \lim_{x \to -2} \frac{x(x+1)}{(x-4)} = \frac{-2(-2+1)}{-2-4} = -\frac{1}{3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/299055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Induction Proof: $\sum_{i=1}^{n+1} i \cdot 2^i = n \cdot 2^{n+2}+2 $ Prove by Mathematical Induction . . .
$$\sum_{i=1}^{n+1} i \cdot 2^i = n \cdot 2^{n+2}+2 $$
for all $n \geq 0$
I tried solving it, but I got stuck near the end . . .
a. Basis Step:
$1\cdot 2^1 = 0\cdot 2^{0+2}+2$
$2 = 2$
b. Inductive Hypothesis
$$\... | Attention: you replaced $\sum_{i=1}^{k+1}2^i=2+2\cdot 2^2+3\cdot 2^3+\ldots+(k+1)2^{k+1}$ by $(k+1)2^{k+1}$, and this is wrong for $k\geq 2$.
The formula holds for $n=0$ as you observed.
Now assume it holds for some $n\geq 0$, i.e.
$$
\sum_{i=1}^{n+1}i2^i=n2^{n+2}+2.
$$
Then
$$
\sum_{i=1}^{n+2}i2^i=\sum_{i=1}^{n+1}i2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/299174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to find the exact value of $ \cos(36^\circ) $? The problem reads as follows:
Noting that $t=\frac{\pi}{5}$ satisfies $3t=\pi-2t$, find the exact value of
$$\cos(36^\circ)$$
it says that you may find useful the following identities:
$$\cos^2 t+\sin^2 t = 1,\\
\sin 2t = 2\sin t\cos t,\\
\sin 3t = 3\sin t - 4\sin... | Here is a way using roots of unity.
We have $\cos 36 = \frac{\omega + \omega^{-1}}{2}$ where $\omega = \text{exp} \left ( \frac{2 i\pi}{10} \right )$.
We have $-w$ is a primitive $5^{th}$ root of unity, since $ -\exp(\frac{2i\pi}{10}) = \exp(i\pi)\exp(\frac{2i\pi}{10}) = \exp(\frac{12i\pi}{10}) = \exp(\frac{2ki\pi}{5})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 1
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The remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$ How to find the remainder of $1^2+3^2+5^2+7^2+\cdots+1013^2$ divided by $8$
| Hint:
$(a_1 + a_2 + \dots + a_n) \mod 8 = [(a_1 \mod 8) + (a_2 \mod 8) + \dots + (a_n \mod 8)] \mod 8$
How may terms are there in the sequence? Take the $n^{th}$ term to be $2n - 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/304041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Max. distance of Normal to ellipse from origin How Can I calculate Maximum Distance of Center of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$ from the Normal.
My Try :: Let $P(a\cos \theta,b\sin \theta)$ be any point on the ellipse. Then equation of Normal at that point is
$ax\sec \theta-by\csc \the... | So, the distance of the normal from the origin $(0,0)$ is $$\left| \frac{a^2-b^2}{\sqrt{(a\sec\theta)^2+(-b\csc\theta)^2}} \right|$$
So, we need to minimize $(a\sec\theta)^2+(-b\csc\theta)^2=a^2\sec^2\theta+b^2\csc^2\theta=f(\theta)$(say)
So, $\frac{df}{d\theta}=a^22\sec\theta\sec\theta\tan\theta+b^22\csc\theta(-\csc\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Values of $a$ for which $(a+4)x^2-2ax+2a-6 <0$ for all $x \in R$ How can we find all values of $a$ for which the inequality $(a+4)x^2-2ax+2a-6 <0$ is satisfied for all $x \in R$?
For the given condition, $D >0$, therefore $ (-2a)^2-4(2a-6)(a+4) >0$. Solving for $a$, I get $(a+6)(a-4) <0$, but the answer is $(-\infty, -... | The quadratic form$(a+4)x^2-2ax+2a-6=\begin{bmatrix} x & 1\end{bmatrix}\begin{bmatrix}a+4& -a \\ -a & 2a-6\end{bmatrix}\begin{bmatrix}x\\ 1\end{bmatrix}$
So if you want the quadratic form to be negative for all $x$, you need a negative definite $\begin{bmatrix}a+4& -a \\ -a & 2a-6\end{bmatrix}$. By Slvester's criteri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Find all integers such that $\frac{n^3-3}{n^2-7}$ is an integer Find all integers such that $\frac{n^3-3}{n^2-7}$ is an integer.
I have no idea how to approach these types of proofs.
But I tried a few things, did not get me anywhere.
$n^3 -3 = an^2-7a$ then $n^3-an^2 = 3-7a$, and hence $n^2(n-a) = 3-7a$
And then I have... | Note that
$$
\frac{n^3-3}{n^2-7} = n + \frac{7n-3}{n^2-7}.
$$
So it suffices to determine when $n^2-7$ divides $7n-3$. It is not hard to show that $|n^2-7| > |7n-3|$ when $|n|>8$; so for those $n$, it is impossible for $n^2-7$ to divide $7n-3$. That leaves only the cases $n=-8,-7,\dots,8$ to check, revealing the two so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$,$F(1)=b$ and $a,b>0$? What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$, $F(1)=b$ and $a,b>0$ ? It seems to be simple generalization of Fibonacci sequence but I can't find closed form for it nor by myself ne... | Note that
$$
\binom{F(n)}{F(n+1)}
=
\begin{bmatrix}
0 & 1\\[6pt]
1 & 1
\end{bmatrix}
\cdot
\binom{F(n-1)}{F(n)}
=
\begin{bmatrix}
0 & 1\\[6pt]
1 & 1
\end{bmatrix}^n
\binom{a}{b}
$$
so the problem essentially reduces to diagonalize the matrix $M=\begin{bmatrix}0&1\\1&1\end{bmatrix}$.
Deno... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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how to prove $\displaystyle \frac{\sin (2n+1)\theta}{\sin \theta} = .... $ How to prove $$ \displaystyle \frac{\sin (2n+1)\theta}{\sin \theta} = (2n+1) \prod_{k=1}^{n}\left(1 - \frac{\sin^2 \theta}{\sin^2 \left( \frac{k\pi }{2n+1} \right ) } \right ) $$
So far, I manage to prove $ \displaystyle \frac{\sin (2n+1)\theta... | Put $z = e^{i\theta}.$ The LHS becomes
$$ \frac{z^{2n+1} - 1/z^{2n+1}}{z-1/z} =
\frac{z^{4n+2} - 1}{z^{2n+2} - z^{2n}} =
\frac{1}{z^{2n}} \frac{z^{4n+2}-1}{z^2-1}.$$
Let $\zeta_k = e^{\frac{2 \pi i k}{4n+2}} = e^{\frac{\pi i k}{2n+1}} $ be the $k$th root of unity.
The RHS is
$$ (2n+1) \prod_{k=1}^n
\left( 1 - \frac{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Ordered pairs $(x,y)$ that satisfy the equation $x^2+y^2 = 2013$ (1) The no. of Integer ordered pair $(x,y)$ that satisfy the equation $x^2+y^2 = 2013$
(2) The no. of Integer ordered pair $(x,y)$ that satisfy the equation $x^2+y^2 = 2014$
My solution for fisrt::
Here R.H.S is a odd quantity means L.H.S must be odd whi... | Note that $2013=3\cdot11\cdot61, 2014=2\cdot19\cdot53$.
$x^2+y^2=2013 \Rightarrow 3 \mid x^2+y^2 \Rightarrow 3\mid x, y \Rightarrow 9 \mid x^2+y^2=2013$, a contradiction.
$x^2+y^2=2014 \Rightarrow 19 \mid x^2+y^2 \Rightarrow 19\mid x, y \Rightarrow 19^2 \mid x^2+y^2=2014$, a contradiction.
Thus there are no integer so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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help for proving an equation by induction For this equation:
$$-1^3+(-3)^3+(-5)^3+\ldots+(-2n-1)^3=(-n-1)^2(-2n^2-4n-1)$$
how can I prove this by induction?
When I set $n = 1$ for the base case I got:
$$-1^3 + (-3)^3 + (-5)^3 + \ldots + (-3)^3 = -28$$
but am having trouble with the following inductive steps
| You’d have had an even simpler base case had you started with $n=0$: $(-1)^3=(-1)^2(-1)$.
For the induction step your induction hypothesis should be that the result is true for some particular $n$, i.e., that
$$(-1)^3+(-3)^3+(-5)^3+\ldots+(-2n-1)^3=(-n-1)^2(-2n^2-4n-1)\;,\tag{1}$$
and you’ll try to prove the correspond... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/312405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$. Which is greater? $\sqrt{n}^{\sqrt{n+1}}$ or $\sqrt{n+1}^\sqrt{n}$
I know that $\sqrt{n}^{\sqrt{n+1}}$ is greater but I tried using induction and I couldn't figure it out. Thanks for the help.
| Hint: take logarithms of both sides and approximate for sufficiently large $n$.
$$\log{\sqrt{n}^{\sqrt{n+1}}} = \frac{1}{2} \sqrt{n+1} \log{n}$$
For sufficiently large $n$, this is approximately
$$-\frac{1}{2}\sqrt{n} \log \left(\frac{1}{n}\right)-\frac{1}{4} \sqrt{\frac{1}{n}} \log
\left(\frac{1}{n}\right)+\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
how to find $\int_{0}^{1}h_n(x)dx?$ I would appreciate if somebody could help me with the following problem:
$$h(x)=\begin{cases}
2x&\left(0\leq x\leq \frac{1}{3}\right)\\
\frac{1}{2}x+\frac{1}{2} &\left(\frac{1}{3}<x\leq 1\right)
\end{cases}$$
let $h_2(x)=h(h(x)), h_3(x)=h_2(h(x)),\cdots, h_{n+1}(x)=h_n(h(x))$
find... | Playing around with the patterns of this recursion, I get the following:
$$h_n(x) = \begin{cases}\\ 2^n x, & 0 < x < \frac{1}{3 \cdot 2^{n-1}}\\2^{n-2 m} x + \left (1-\frac{1}{2^m}\right ),&\frac{1}{3 \cdot 2^{n-m}}<x<\frac{1}{3 \cdot 2^{n-m-1}} \\ \frac{1}{2^n} x +\left (1-\frac{1}{2^n}\right ),&\frac{1}{3}<x<1 \end{c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/320068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
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Evaluate $\frac{1}{2\pi}\int _{-\pi}^\pi \frac{e^{-i\varphi}}{1-k\cos (\varphi)} \, \mathrm{d}\varphi$ I couldn't evaluate this integral. Could you please help me?
$$\frac{1}{2\pi}\int _{-\pi}^\pi \frac{e^{-i\varphi}}{1-k\cos (\varphi)} \, \mathrm{d}\varphi,\text{ where $k$ is a constant}$$
| Following up on Yimin's suggestion:
$$
\int_{-\pi}^\pi \frac{e^{-i\varphi}}{1-k\cos\varphi} \, d\varphi = \int_{-\pi}^\pi \frac{\cos\varphi}{1-k\cos\varphi} \, d\varphi - i\int_{-\pi}^\pi \frac{\sin\varphi}{1-k\cos\varphi} \, d\varphi.
$$
Since the second integral is that of an odd function over an interval symmetric ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/325683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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What would the space under the image of this function look like? The function is as follows: $$\phi(x,y,z) = \left(\frac{yz}{x^2+y^2+z^2},\frac{zx}{x^2+y^2+z^2}\right)$$
And when $z=0$,the function becomes $ \; \phi(x,y,0)=\frac{xy}{x^2+y^2}$. Some properties of this function are that for any point $ \; p = (x,y,z) \in... | I claim that the image equals the closed ball $B$ with radius $1/2$ and center $(0,0)$.
For $\mathrm{im}(\phi) \subseteq B$, observe that
$$\left(\frac{yz}{x^2+y^2+z^2}\right)^2 + \left(\frac{zx}{x^2+y^2+z^2}\right)^2=\frac{z^2 (y^2+x^2)}{(x^2+y^2+z^2)^2} \leq \frac{1}{4}$$
since for all $a,b,c \geq 0$ we have $(a+b+c)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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trigonometric equation involving two variables Show that the equation $\sec^2\theta = \frac{4xy}{(x+y)^2}$ is only possible when $x =y$.
Can we write it this way : $$\cos^2\theta = \frac{(x+y)^2}{4xy}$$ and $$-1 \leq \cos\theta \leq 1 \Longrightarrow -1 \leq \frac{(x+y)^2}{4xy} \leq 1 $$
or $-4xy \leq (x+y)^2 \leq 4xy ... | $$\sec^2\theta = \frac{4xy}{(x+y)^2}$$
$$\implies \tan^2\theta=\sec^2\theta-1$$
$$=\frac{4xy}{(x+y)^2}-1=\frac{4xy-(x+y)^2}{(x+y)^2}=-\frac{(x-y)^2}{(x+y)^2}=-\left(\frac{x-y}{x+y}\right)^2$$
As $\theta$ is real, $\tan^2\theta\ge0$
$\implies -\left(\frac{x-y}{x+y}\right)^2\ge0$
$\implies \left(\frac{x-y}{x+y}\right)^2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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A question about an equilateral triangle Suppose that $\triangle ABC$ is an equilateral triangle. Let $D$ be a point inside the triangle so that $\overline{DA}=13$, $\overline{DB}=12$, and $\overline{DC}=5$. Find the length of $\overline{AB}$.
|
Let s be the side of the equilateral triangle
Using cosine formula for the triangle
$$\frac{13^2 + 5^2 - s^2}{2.5.13} = \cos\alpha\tag1$$
$$\frac{12^2 + 5^2 - s^2}{2.5.12} = \cos\beta\tag2$$
$$\frac{12^2 + 13^2 - s^2}{2.13.12} = \cos\gamma\tag3$$
Now we have $\gamma + \alpha + \beta = \pi$
$$\cos\gamma = \cos(\pi - \a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/330333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$? How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$
I found the above interesting identity in the book $\bf \pi$ Unleashed.
Does anyone knows how to prove it?
Thanks.
| There is a simple way to compute the sum. Note
$$ \sum_{k=-\infty}^\infty\frac{\sin^2k}{k^2}=1+2\sum_{k=1}^\infty \frac{\sin^2k}{k^2}=1+\sum_{k=1}^\infty\frac{1-\cos(2k)}{k^2}=1+\frac{\pi^2}{6}-\sum_{k=1}^\infty\frac{\cos(2k)}{k^2}.$$
Letting $x=e^{2\theta i}$ in $\sum_{k=1}^\infty\frac{x^k}{k}=-\ln(1-x)$ gives us
$$ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "77",
"answer_count": 7,
"answer_id": 6
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Area between 3 curves I need to calculate an area between those 3 curves.
I have no idea what area should I calculate (because of y=-x-1)
Regards.
| It is very simple(this is school example?).
Simply compute $\int_a^0 [(x-1)^3 - 3\cdot x - 5] {d}x$ where a is intesect of $(x-1)^3$ with $3\cdot x - 5$(if in Wolfram|Alpha enter "$[solve (x-1)^3 = 3\cdot x - 5][1]$" you get $x = -1$ and $x = 2$, so $a = -1$): $\int_{-1}^0 [(x-1)^3 - 3\cdot x + 5] {d}x = \left.{{x}\o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Help with inequality please Once again I have come across an olympiad-type problem which probably requires some sort of insight even though it looks simple. The question is as follows:
Let $a$, $b$ and $c$ be positive real numbers. Prove that:
$(a+b)(b+c)(c+a)$ $\geqslant$ $8(a+b-c)(b+c-a)(c+a-b)$
I have tried to multi... | Given $\quad(a+b)(b+c)(c+a) \ge 8(a+b-c)(b+c-a)(c+a-b)\quad$ show $x\ge y$
$$x=(a+b)(b+c)(c+a) \quad=a^2 b + a^2 c + a b^2 + 2 a b c + a c^2 + b^2 c + b c^2$$
$$y=8(a+b-c)(b+c-a)(c+a-b) \\ =
-8 a^3 + 8 a^2 b + 8 a^2 c + 8 a b^2 - 16 a b c + 8 a c^2 - 8 b^3 + 8 b^2 c + 8 b c^2 - 8 c^3$$
We negate the RHS and subtract... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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How to find the $f\left(x_i\right)$ for midpoint Riemann sum
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\mbox{Time} & \mbox{9am} & \mbox{10:30 am} & \mbox{Noon} & \mbox{1:30 pm} & \mbox{3 pm} & \mbox{4:30 pm} & \mbox{6 pm} & \mbox{7:30 pm} & \mbox{9 pm} \\
\hline
t & 0 & 1.5 & 3 & 4.5 & 6 & 7.5 & 9 & 10.5 & 12 \\
\h... | For $n$ points from $a$ to $b$,
$x_i = a + \frac{b-a}{n}(i-\frac1{2})$.
Note that $x_1 = a + \frac1{2}\frac{b-a}{n}$
is just to the right of $a$,
and $x_n = a+(n-\frac1{2})\frac{b-a}{n}
= b - \frac1{2}\frac{b-a}{n}$
is just to the left of $b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/335332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Find all solutions of $7x^2 \equiv 1 \pmod {17}$ Find all solutions of $7x^2 \equiv 1 \pmod {17}$
I found out all the primitive root of $U_{17}$ to be : $\{3,5,6,7,10,12,14\}$.
To continue with the computation, I think i need to use the theorem which is $x^2 \equiv1\pmod{n}$. But Im not sure how to connect it with the... | $$7x^2 \equiv 1 \pmod{17} \implies x^2 \equiv 7^{-1} \pmod{17} \implies x^2 \equiv 5 \pmod{17}$$
When $(x,17) = 1$ from little-Fermat, we have
$$x^{16} \equiv 1 \pmod{17}$$
But we also need $x^2 \equiv 5 \pmod{17}$. This means $$x^{16} = {x^2}^8 \equiv 5^8 \pmod{17} \equiv (25)^4 \pmod{17} \equiv 8^4 \pmod{17} \equiv (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/336332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Find all solutions of $x^8 \equiv 5 \pmod {3\cdot 11}$ Find all solutions of $x^8 \equiv 5 \pmod {3\cdot 11}$
Im not sure should I use primitive root or quadratic residue.
For primitive root, $U_{33} = \{1,2,4,5,7,8,10,13,14,16,17,19,20,23,25,26,28,29,31,32\}$
Thus $\phi_{(3\cdot11)}=20=2^2\times{5}$
But it takes way ... | Hint $\rm\ mod\ 3\!:\ x\not\equiv 0\:\Rightarrow\:x\equiv \pm 1\:\Rightarrow\:x^2\equiv 1\:\Rightarrow\:x^{2n}\!\equiv 1\not\equiv 5$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/336378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Showing series expansion How would I show $$\frac{\sinh z}{z^2}=\frac{1}{z}+\sum_0^\infty \frac{z^{2n+1}}{(2n+3)!}$$
I have $\sinh z=\sum_0^\infty\frac{z^{2n+1}}{(2n+1)!}$ and if I multiply by $\frac{1}{z^2}$ then $\frac{1}{z^2}\sinh z=\sum_0^\infty \frac{1}{z^2}\frac{z^{2n+1}}{(2n+1)!}$ but I how would I would factor... | Write out a few terms to see what is going on.
$$\begin{align}
\sinh z &= \frac{z}{1!}+\frac{z^3}{3!} + \frac{z^5}{5!}+\dots
\\
\frac{\sinh z}{z^2} &= \frac{1}{1! z}+\frac{z}{3!} + \frac{z^3}{5!}+\dots
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/337207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Trigonometric Functions And Identities Question . If $A$ , $B$ and $C$ are the angles of a triangle , I have to show that :
$ \tan^2 \cfrac{A}{2} + \tan^2 \cfrac{B}{2} + \tan^2 \cfrac{C}{2} \ge 1 $ .
I had only arrived at $A+B+C = \pi $ , thus $\cfrac{A}{2} + \cfrac{B}{2} + \cfrac{C}{2} = \cfrac{\pi}{2} $
What to do... | First note that $\dfrac{A}2, \dfrac{B}2, \dfrac{C}2 \in (0, \pi/2)$. Now consider the function $f(x) = \tan^2(x)$. For $x \in (0,\pi/2)$, $f(x)$ is convex. Hence, we have
$$\dfrac{f(x_1) + f(x_2) + f(x_3)}3 \geq f \left(\dfrac{x_1 + x_2 + x_3}3 \right)$$
Take $x_1 = \dfrac{A}2$, $x_2 = \dfrac{B}2$ and $x_3 = \dfrac{C}2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Find the limit (if it exists): $\lim_{(x,y,x)\rightarrow(0,0,0)}\frac{xyz}{x^2+y^2+z^2}$ Find the limit (if it even exists). If not, prove it doesn't exist.
$$\lim_{(x,y,x)\rightarrow(0,0,0)}\frac{xyz}{x^2+y^2+z^2}$$
| If you use the facts
$$ |x|=\sqrt{x^2}\leq \sqrt{x^2+y^2+z^2}, \quad |y|=\sqrt{y^2}\leq \sqrt{x^2+y^2+z^2},\quad |z|=\sqrt{z^2}\leq \sqrt{x^2+y^2+z^2}, $$
then the problem falls apart as
$$ \Bigg|\frac{xyz}{x^2+y^2+z^2}\Bigg|= \frac{|xyz|}{x^2+y^2+z^2}= \frac{|x||y||z|}{x^2+y^2+z^2} $$
$$ \leq \frac{(x^2+y^2+z^2)^{3/2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Evaluate $\sum_{n=1}^{\infty}\frac{1}{n^3+3 n^2+2 n}$ Summing this series from $0$ to $\infty$, the result is $\frac{1}{4}$. I tried a lot, but I could not get this result. I think it´s wrong.
Can anybody help me?
| Hints - Partial Fractions:
$$n^3+3n^2+2n=n(n+1)(n+2)\Longrightarrow$$
$$\frac{1}{n^3+3n^2+2n}=\frac{1}{n(n+1)(n+2)}=\frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2(n+2)}\Longrightarrow$$
$$\sum_{n=1}^\infty\frac{1}{n^3+3n^2+2n}=\left(\frac{1}{2}-\frac{1}{2}+\frac{1}{6}\right)+\left(\frac{1}{4}-\frac{1}{3}+\frac{1}{8}\right)=\ldo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/341650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 0
} |
Exponential and logarithmic series: Find the sum of $2^2 + 3^2/ 2!+4^2/3!+...$ to infinity
Find the sum of the following series:
$ 2^2 + 3^2/2! + 4^2/3! + ...$ to infinity
The answer is given as $5e$ but I got it as $5e+1$
$T_n = 1/(n-2)! +3/(n-1)! + 1/(n)! $ for $n \ge 2$
and $T_1+T_2 + ... $ to infinity = $4 + e... | The answer is $5e-1$. There might be some confusion with the initial values of the sum.
By letting $T_n = \frac{1}{(n-2)!} + \frac{3}{(n-1)!} + \frac{1}{n!}$, as you did:
$S = 2^2 + T_2 + T_3 + \dots = 4 + (\frac{1}{0!} + \frac{3}{1!} + \frac{1}{2!}) + (\frac{1}{1!} + \frac{3}{2!} + \frac{1}{3!}) + (\frac{1}{2!} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/348061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Let $r,s,t$ be the roots of the equation $ x^3 - 6x^2 + 5x + 1$. What is the value of $(2-r)(2-s)(2-t)$? Let $r,s,t$ be the roots of the equation $ x^3 - 6x^2 + 5x + 1$. What is the value of $(2-r)(2-s)(2-t)$?
The question is mentioned in my math olympiad. Please explain how to solve the problem. I have factorised the ... | If $c$ is a root of some polynomial, then $x - c$ is a a factor of that polynomial. Furthermore, if the polynomial has degree $n$, then it has at most $n$ roots. So $$x^3 - 6x^2 + 5x + 1 = (x - r)(x - s)(x - t)$$ Thus,
\begin{align*}
(2 - r)(2 - s)(2 - t) &= 2^3 -6(2^2) + 5(2) + 1 \\
&= 8 - 6(4) + 10 + 1 \\
&= 8 - 24 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/351650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 3
} |
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Fractions in Questions and Answers
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