Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
A simple conditional expectation problem $X, Y$ iid uniform random variables on $[0,1]$
$$Z =
\left\{
\begin{aligned}
X+Y \quad&\text{ if } X>\frac{1}{2}
\\
\frac{1}{2} + Y \quad & \text{ if } X\leq\frac{1}{2}
\end{aligned}
\right.$$
The question is $E\{Z|Z\leq 1\}= ?$
I tried $\displaystyle \int_0^1 E\{Z|Z = z... | For $x \in (1/2,1]$, let
$$
F_{Z|Z \leq 1}(x) = {\rm P}(Z \leq x | Z \leq 1) = \frac{{{\rm P}(Z \le x)}}{{{\rm P}(Z \le 1)}},
$$
and let
$$
f_{Z|Z \leq 1}(x) = \frac{{\rm d}}{{{\rm d}x}}F_{Z|Z \le 1} (x) = \frac{1}{{{\rm P}(Z \le 1)}}\frac{{\rm d}}{{{\rm d}x}}{\rm P}(Z \le x).
$$
Then,
$$
{\rm E}[Z|Z \le 1] = \int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/52317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proving that $ 30 \mid ab(a^2+b^2)(a^2-b^2)$ How can I prove that $30 \mid ab(a^2+b^2)(a^2-b^2)$ without using $a,b$ congruent modulo $5$ and then
$a,b$ congruent modulo $6$ (for example) to show respectively that $5 \mid ab(a^2+b^2)(a^2-b^2)$ and
$6 \mid ab(a^2+b^2)(a^2-b^2)$?
Indeed this method implies studying nu... | Hint: First write $f(a,b)=ab(a^2+b^2)(a^2-b^2)=a^5b-b^5a$. Recall Fermat's Little Theorem which states $x^{p-1}\equiv 1 \pmod{p}$ for $x$ not divisible by $p$.
Assume $5$ does not divide either of $a,b$. (otherwise it follows automatically that $5|f(a,b)$) Then
$$a^5b-b^5a\equiv ab-ba\equiv 0\pmod{5}$$ and we see tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/53135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
Why does this expression equal $\pi$? I noticed that the following expression equals $\pi$ and I was curious as to why. Is it just a coincidence, or is there a meaningful explanation?
$$\int_{-\infty}^\infty\frac{1}{x^2+1}~dx=\pi$$
| Here is another approach, which avoids taking $\arctan(\infty)$, etc.: let $u = \frac{1}{x}$, so that $\frac{du}{dx} = \frac{-1}{x^2}.$ Then we obtain $\int_{1}^{\infty} \frac{1}{1+u^{2}} du = \int_{0}^{1} \frac{1}{1+x^2}dx$. Hence $\int_{- \infty}^{\infty} \frac{1}{1+x^2} dx = 4\int_{0}^{1} \frac{1}{1+x^2}dx = 4 \arct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/54414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 2
} |
Formula for $1^2+2^2+3^2+...+n^2$ In example to get formula for $1^2+2^2+3^2+...+n^2$ they express $f(n)$ as:
$$f(n)=an^3+bn^2+cn+d$$ also known that $f(0)=0$, $f(1)=1$, $f(2)=5$ and $f(3)=14$
Then this values are inserted into function, we get system of equations solve them and get a,b,c,d coefficients and we get that... | It is a consequence of the following algebraic identity
$$1+2^{2}+3^{2}+\ldots +n^{2}=\frac{1}{3}\left( n^{3}+3n^{2}+3n+1\right) -
\frac{1}{3}n-\frac{1}{2}(n^2+n)-\frac{1}{3}.$$
$$\tag{1}$$
The RHS is a cubic function of $n$: $\frac{1}{3}n^{3}+\frac{1}{2}n^{2}+\frac{1}{6}n$.
Proof. From the algebraic identity
$$\left( ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/59175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ show that $x=-c/b$ when $a=0$ OK, this one has me stumped. Given that the solution for $ax^2+bx+c =0$ $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\qquad(*)$$
How would you show using $(*)$ that $x=-c/b$ when $a=0$
(Please dont use $a=0$ hence, $bx+c=0$.)
| You don't. The quadratic formula only works if $a\neq 0$ (remember, if you try dividing by $0$, the universe explodes).
If $a=0$, then the equation is
$$0x^2 + bx + c = 0$$
or equivalently,
$$bx + c = 0.$$
So, solve that equation for $x$. If $b=0$, then it's always false if $c\neq 0$, and always true if $c=0$.
And if $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/60792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
How to prove that the Binet formula gives the terms of the Fibonacci Sequence? This formula provides the $n$th term in the Fibonacci Sequence, and is defined using the recurrence formula: $u_n = u_{n − 1} + u_{n − 2}$, for $n > 1$, where $u_0 = 0$ and $u_1 = 1$.
Show that
$$u_n = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{... | Let
$$A=\begin{pmatrix}
1 & 1 \\ 1 & 0
\end{pmatrix}$$
$$M_n =
\begin{pmatrix}
F_{n+1} & F_n \\ F_n & F_{n-1}
\end{pmatrix}$$
By induction, you can show that $M_n=A^n$. Now, the eigenvalues of $A$ satisfy $\lambda^2-\lambda-1=0$. Let them be $\lambda_1$ and $\lambda_2$.
$A$ can be diagonalized and written as $A=SBS... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/65011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 7,
"answer_id": 1
} |
Just checking: sine series for $x^2$ Is the Fourier sine series for $x^2$ equal to $\sum {2\pi\over 2m+1}-{8\over (2m+1)^3\pi} \sin ((2m+1)x)$? (just want to check that those multiple steps of intergation by parts did not slip me up). Thanks.
| The Fourier sine series is $x^2 = \sum_{n=1}^\infty a_n \sin( n x)$, where $a_n = \frac{2}{\pi} \int_0^\pi x^2 \sin( n x) \mathrm{d} x$. Now
$$
\begin{eqnarray}
\frac{\pi}{2} a_n &=& \left. (-\frac{x^2}{n} \cos(n x)) \right\vert_{0}^{\pi} + \int_0^\pi ( \frac{2}{n} x) \cos(n x) \mathrm{d} x \\
&=& \left. (-\frac{x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/66347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\log_{12} 2=m$ what's $\log_6 16$ in function of $m$? Given $\log_{12} 2=m$ what's $\log_6 16$ in function of $m$?
$\log_6 16 = \dfrac{\log_{12} 16}{\log_{12} 6}$
$\dfrac{\log_{12} 2^4}{\log_{12} 6}$
$\dfrac{4\log_{12} 2}{\log_{12} 6}$
$\dfrac{4\log_{12} 2}{\log_{12} 2+\log_{12} 3}$
$\dfrac{4m}{m+\log_{12} 3}$
And thi... | Writing everything without logarithms:
$$
\begin{array}{ccc}
12^m=2&\therefore&3=2^{1/m-2}\\
6^x=16&\therefore&3=2^{4/x-1}
\end{array}
$$
Thus we get
$$
\begin{array}{ccc}
1/m-2=4/x-1&\therefore&x=\frac{4m}{1-m}
\end{array}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/66405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Determine a point $$\text{ABC- triangle:} A(4,2); B(-2,1);C(3,-2)$$
Find a D point so this equality is true:
$$5\vec{AD}=2\vec{AB}-3\vec{AC}$$
| $$\text{The given vectors } \overrightarrow{AB}=B-A\text{ and }\overrightarrow{AC}=C-A \text{ and the solution }D=A+\overrightarrow{AD}$$
Let $(x,y)$ be the coordinates of $D$. The equation
$$5\overrightarrow{AD}=2\overrightarrow{AB}-3\overrightarrow{AC}\tag{0}$$
means that
$$5(x-4,y-2)=2(-2-4,1-2)-3(3-4,-2-2),\tag{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/66671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Diverging sequence I can't understand diverging sequences. How can I prove that $a_n=1/n^2-\sqrt{n}$ is divering? Where to start? What picture should I have in my mind? I tried to use $\exists z \forall n^* \exists n\ge n^*: |a_n-A|\ge z$, but how should I see this? And how can I solve the question with this property?
| A convergent sequence is also a Cauchy sequence. If you can find a constant $\epsilon>0$ so that for all $N>0$, there are $n>N$ and $m>N$ so that $|a_n-a_m|\ge\epsilon$, then $\{a_n\}$ is a divergent sequence.
In the case of $a_n=\frac{1}{n^2}-\sqrt{n}$, we can let $\epsilon=1$ and for a given $N>0$, let $n=N+1$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/67476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Finding sum form for a particular recursive function Consider a finite sequence of zeros and ones of length $3n$, with $n$ an integer. We write an element of this sequence as $a_i$. How many sequences are there such that there exists an integer $k$, $0<k\le n$, such that $\sum^{3k}_{j=1}a_j=2k$? Here is what I have as ... | If $x_n$ was $\binom{3n}{2n}+5x_{n-1}$, which it isn’t, we would have
$$
x_n=\sum_{k=1}^{n} 5^{(n-k)} \binom{3k}{2k}
$$
Mathematica gives the following “closed form”:
$$
x_n=-\frac{1}{5} \binom{3 n+3}{2 n+2} \, _3F_2\left(1,n+\frac{4}{3},n+\frac{5}{3};n+\frac{3}{2},n+2;\frac{27}{20}\right)
-\alpha \;5^n $$
where ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Complex solutions for Fermat-Catalan conjecture The Fermat-Catalan conjecture is that $a^m + b^n = c^k$ has only a finite number of solutions when $a, b, c$ are positive coprime integers, and $m,n,k$ are positive integers satisfying $\frac{1}{m} + \frac{1}{n} +\frac{1}{k} <1$. There are currently only 10 solutions know... | Here is a small portion of what I have found:
$i^{4 m}+2^3=3^2$ where integer $m \ge 2$, i.e. the smallest is $i^8+2^3=3^2$
$(1+i)^2=i^{4 m+1}+i^{4 n+1}$ where integer $m \ge 1$, $n \ge 1$, i.e. the smallest is $(1+i)^2=i^5+i^5$
$(78-78 i)^2=(23 i)^3+i^{4 m+3}$ where integer $m \ge 1$, i.e. the smallest is $(78-78 i)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/69291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 2,
"answer_id": 1
} |
Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ Given the geometric series:
$1 + x^2 + x^4 + x^6 + x^8 + \cdots$
We can recast it as:
$S = 1 + x^2 \, (1 + x^2 + x^4 + x^6 + x^8 + \cdots)$, where $S = 1 + x^2 + x^4 + x^6 + x^8 + \cdots$.
This recasting is possible only because there is an infin... | There is a finite version of which the expression you have is the limit.
Suppose $S=1+x^2+x^4+x^6+x^8$, then we can put
$S+x^{10}=1+x^2(1+x^2+x^4+x^6+x^8)=1+x^2S$
And obviously this can be taken as far as you like, so you can replace 10 with 10,000 if you choose. If the absolute value of $x$ is less than 1, this extra ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/70048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Solving a recurrence using substitutions I have to solve this recurrence using substitutions:
$(n+1)(n-2)a_n=n(n^2-n-1)a_{n-1}-(n-1)^3a_{n-2}$ with $a_2=a_3=1$.
The only useful substitution that I see is $b_n=(n+1)a_n$, but I don't know how to go on, could you help me please?
| it seems that I solved. $$(n-2)b_n=(n^2-n-1)b_{n-1}-(n-1)^2b_{n-2}$$
So divide it by $n-1$ then we get $$\begin{align*}
\left(1-\frac{1}{n-1}\right)b_n&=\left(n-\frac{1}{n-1}\right)b_{n-1}-(n-1)b_{n-2}\\
&=\left(n-1+1-\frac{1}{n-1}\right)b_{n-1}-(n-1)b_{n-2}\\
&=(n-1)(b_{n-1}-b_{n-2})+b_{n-1}\left(1-\frac{1}{n-1}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to sum this series for $\pi/2$ directly? The sum of the series
$$
\frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1}
$$
can be derived by accelerating the Gregory Series
$$
\frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2}
$$
using Euler's Series Transformation. Mathematica is able to sum $(1)$, ... | Notice that for $c_k = \frac{k!}{(2k+1)!!}$ the ratio of successive terms $\frac{c_{k+1}}{c_k} = \frac{k+1}{2k +3} = \frac{1}{2} \frac{k+1}{k+3/2}$.
This means that the series is hypergeometric with the value ${}_2 F_1(1, 1, \frac{3}{2}, \frac{1}{2})$.
This particular Gaussian hypergeometric is elementary:
$$
{}_2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "111",
"answer_count": 5,
"answer_id": 4
} |
Prove the identity $ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$ $$ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$$
Class themes are: Generating functions and formal power series.
| Ok, here is an approach with generating functions. Let
$$
g_1(z) = \sum_{s=0}^\infty \binom{p+s}{s} z^s = \frac{1}{\left(1-z\right)^{p+1}}
$$
$$
g_2(z) = \sum_{s=0}^\infty \binom{2p+m}{s} z^s = \left(1+z\right)^{m+2p}
$$
Now
$$ \begin{eqnarray}
\sum_{s=0}^\infty \binom{p+s}{s} \binom{2p+m}{2p+2s} &=& \sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 1
} |
Evaluate: $\lim_{n\to\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac1{\sqrt k}\right)$ How to find $\lim\limits_{n\to\infty}\left({2\sqrt n}-\sum\limits_{k=1}^n\frac1{\sqrt k}\right)$ ?
And generally does the limit of the integral of $f(x)$ minus the sum of $f(x)$ exist?
How to prove that and find the limit?
| Use $\sqrt{n} = \sum_{k=1}^n \left( \sqrt{k} - \sqrt{k-1} \right)$, then
$$
\begin{eqnarray}
2 \sqrt{n} - \sum_{k=1}^n \frac{1}{\sqrt{k}} &=& \sum_{k=1}^n \left( 2 \sqrt{k} - 2 \sqrt{k-1} - \frac{1}{\sqrt{k}} \right) = \sum_{k=1}^n \frac{1}{\sqrt{k}} \left( \sqrt{k}-\sqrt{k-1} \right)^2\\
&=& \sum_{k=1}^n \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 3,
"answer_id": 2
} |
Converting multiplying fractions to sum of fractions I have the next fraction: $$\frac{1}{x^3-1}.$$
I want to convert it to sum of fractions (meaning $1/(a+b)$).
So I changed it to:
$$\frac{1}{(x-1)(x^2+x+1)}.$$
but now I dont know the next step. Any idea?
Thanks.
| $x^2+x+1=(x-a)(x-\bar{a})$ where $a=\exp(\frac{2\pi i}{3})=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$, so
$$
\frac{1}{x^3-1}=\frac{1}{(x-1)(x-a)(x-\bar{a})}\tag{1}
$$
and then you can use partial fractions on $(1)$ to get
$$
\frac{1}{x^3-1}=\frac{1}{3}\left(\frac{1}{x-1}+\frac{a}{x-a}+\frac{\bar{a}}{x-\bar{a}}\right)\tag{2}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Why $\frac{1}{\cos(\sin^{-1}(x))}=\frac{1}{\sqrt{1-x^2}}$? This is the inverse function of sin. Why is $\cos(\sin^{-1}x)=\sqrt{1-x^2}$?
Thanks a lot.
| Draw a right angled triangle a marked angle $\theta $. We can scale the triangle to set the hypotenuse equal to 1. Label the side opposite $\theta $ by some length $x$, then the adjacent side has length $\sqrt{1-x^2}.$
$\phi = \sin^{-1} x $ denotes the angle such that $\sin \phi = x $. So in the triangle, $ \sin^{-1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Determine if the equation is valid/true The equation is:
$$\log_b \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} = 2\log_b(\sqrt{3}+\sqrt{2}).$$
I can get as far as:
$$\log_b(\sqrt{3}+\sqrt{2}) - \log_b(\sqrt{3}-\sqrt{2}) = 2\log_b(\sqrt{3}+\sqrt{2})$$
Which looks almost too simple, but I can't get the signs to match up r... | $$\log_b(\sqrt{3}+\sqrt{2}) - \log_b(\sqrt{3}-\sqrt{2}) = 2\log_b(\sqrt{3}+\sqrt{2})$$
is equivalent to
$$\log_b(\sqrt{3}+\sqrt{2}) = \log_b(\sqrt{3}-\sqrt{2}) + 2\log_b(\sqrt{3}+\sqrt{2})$$
which is equivalent to
$$\log_b(\sqrt{3}+\sqrt{2}) = \log_b((\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})^2)$$
which you can multip... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Prove equations in modular arithmetic Prove or disprove the following statement in modular arithmetic.
*
*If $a\equiv b \mod m$, then $ a^2\equiv b^2 \mod m$
*If $a\equiv b \mod m$, then $a^2\equiv b^2 \mod m^2$
*If $a^2\equiv b^2\mod m^2$, then $a\equiv b\mod m$
My proofs.
*
*$$ a\equiv b \mod m \implies (... | For Question 3, you are told that $m^2$ divides $a^2-b^2$, and are asked whether (necessarily) $m$ divides $a-b$.
Note that $a^2-b^2=(a-b)(a+b)$. Maybe the "factor" $m^2$ comes in whole or in large part from $a+b$, not from $a-b$. Let's see whether we can find an example. Let $m=3$, so $m^2=9$. Make $a+b$ divisible b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/89363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Maxima of bivariate function [1] Is there an easy way to formally prove that,
$$
2xy^{2} +2x^{2} y-2x^{2} y^{2} -4xy+x+y\ge -x^{4} -y^{4} +2x^{3} +2y^{3} -2x^{2} -2y^{2} +x+y$$
$${0<x,y<1}$$
without resorting to checking partial derivatives of the quotient formed by the two sides, and finding local maxima?
[2] Similar... | Certainly, there is no need for taking the quotient, since $a \ge b \Leftrightarrow \min \{a-b\} \ge 0$.
Here's a cool trick called the S.O.S. (sum of squares) method. The idea is to try and factor out $(x-y)^2$:
$$\begin{align}
LHS-RHS &=(x^4+y^4-2x^2y^2)-2(x^3+y^3-x^2y-xy^2)+2(x^2+y^2-2xy)\\
&=(x^2-y^2)^2-2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is my trig result unique? I recently determined that for all integers $a$ and $b$ such that $a\neq b$ and $b\neq 0$,
$$
\arctan\left(\frac{a}{b}\right) + \frac{\pi}{4} = \arctan\left(\frac{b+a}{b-a}\right)
$$
This implies that 45 degrees away from any angle with a rational value for tangent lies another angle with a ... | As written, the formula is not true: the values of $\arctan(x)$ are always between $-\frac{\pi}2$ and $\frac{\pi}{2}$. Pick a rational number $\frac{a}{b}$ with $\frac{\pi}{4}\lt \frac{a}{b}\lt \frac{\pi}{2}$. For example, $a=11$, $b=10$. Then the left hand side,
$$\arctan\left(\frac{11}{10}\right)+\frac{\pi}{4}\approx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
Adding a different constant to numerator and denominator Suppose that $a$ is less than $b$ , $c$ is less than $d$.
What is the relation between $\dfrac{a}{b}$ and $\dfrac{a+c}{b+d}$? Is $\dfrac{a}{b}$ less than, greater than or equal to $\dfrac{a+c}{b+d}$?
| Note that if $b$ and $d$ have the same sign, then
$$
\frac{a}{b}-\frac{a+c}{b+d}=\frac{ad-bc}{b(b+d)}
$$
and
$$
\frac{a+c}{b+d}-\frac{c}{d}=\frac{ad-bc}{d(b+d)}
$$
also have the same sign.
Therefore, if $b$ and $d$ have the same sign, then $\dfrac{a+c}{b+d}$ is between $\dfrac{a}{b}$ and $\dfrac{c}{d}$.
Comment: As... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
What is the meaning of $\mathbf{Q}(\sqrt{2},\sqrt{3})$ I know that : $\mathbf{Q}(\sqrt{2}) = \mathbf{Q}+ \sqrt{2} \mathbf{Q}$ , but then what is $\mathbf{Q}(\sqrt{2},\sqrt{3})$?
| $\mathbf{Q}(\sqrt{2},\sqrt{3})$ means $\mathbf{Q}+\sqrt{2}\mathbf{Q}+\sqrt{3}\mathbf{Q}+\sqrt{6}\mathbf{Q}$, or in other words
$$\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\mid a,b,c,d\in\mathbf{Q}\}.$$
Be careful though. For example, $\mathbf{Q}(\sqrt{2},\sqrt[4]{2})=\mathbf{Q}(\sqrt[4]{2})=\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Is every Mersenne prime of the form : $x^2+3 \cdot y^2$? How to prove or disprove following statement :
Conjecture :
Every Mersenne prime number can be uniquely written in the form : $x^2+3 \cdot y^2$ ,
where $\gcd(x,y)=1$ and $x,y \geq 0$
Since $M_p$ is an odd number it follows that : $M_p \equiv 1 \pmod 2$
Accordin... | It is a theorem that may have been first proved by Euler (but was known to Fermat) that every prime $p$ of the form $6k+1$ can be expressed in the form $p=x^2+3y^2$, where $x$ and $y$ are integers.
This representability question has been discussed on StackExchange. Please see here for a compact complete proof by Ewan D... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is $\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$? $$\sin^3 x=\frac{3}{4}\sin x - \frac{1}{4}\sin 3x$$
Is there any formula that tells this or why is it like that?
| de Moivre's formula says
$$
\begin{align}
\cos(3x)+i\sin(3x)
&=(\cos(x)+i\sin(x))^3\\
&=\left(\cos^3(x)-3\cos(x)\sin^2(x)\right)+i\left(3\cos^2(x)\sin(x)-\sin^3(x)\right)\\
&=\left(4\cos^3(x)-3\cos(x)\right)+i\left(3\sin(x)-4\sin^3(x)\right)\tag{1}
\end{align}
$$
Therefore,
$$
\begin{align}
\cos(3x)&=4\cos^3(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 8,
"answer_id": 3
} |
Convergence of the next series I'm trying to determine the convergence of this series:
$$\sum \limits_{n=1}^\infty\left(\frac12·\frac34·\frac56·...\frac{2n-3}{2n-2}·\frac{2n-1}{2n}\right)^a$$
I've tried using D'Alambert criteria for solving it.
$$\lim_{n->\infty}\frac{(\frac12·\frac34·\frac56·...\frac{2n-3}{2n-2}·\frac... | Let $$a_n={1\over2}\cdot{3\over4}\cdot{5\over6}\cdot\ \cdots\ \cdot {{2n-3}\over{2n-2}}\cdot{{2n-1}\over{2n}}.$$
Note that $a_{n+1}=a_n\cdot {2n +1\over2(n+1)}$.
We will show that for all $n$:
$$
{1\over\sqrt{ 4n}} \le a_n\le {1\over\sqrt{ 2n+1}}.
$$
Having established this, it will follow by co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$ $n = 4k + 3 $
We start by letting $a \not\equiv 0\pmod n$ $\Rightarrow$ $a \equiv k\pmod n$ .
$\Rightarrow$ $a^{4k+2} \equiv 1\pmod n$
Now, I know that the contradiction will arrive from the fact that if we can show $a^2 \... | Hint: If $p \equiv 3 \pmod{4}$, then $x$ is a quadratic residue $\bmod{p}$ if and only if $-x$ is a quadratic nonresidue $\bmod{p}$.
Edit, to be more direct:
Lemma:
Suppose that $p$ is an odd prime, and assume that $x$ is a nonzero quadratic residue $\bmod{p}$. Then, $-x$ is a quadratic residue $\bmod{p}$ if and on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
proof that if $AB=BA$ matrix $A$ must be $\lambda E$ Let $A \in Mat(2\times 2, \mathbb{Q})$ be a matrix with $AB = BA$ for all matrices $B \in Mat(2\times 2, \mathbb{Q})$.
Show that there exists a $\lambda \in \mathbb{Q}$ so that $A = \lambda E_2$.
Let $E_{ij}$ be the matrix with all entries $0$ except $e_{ij} = 1$.
$... | Take $B$ to be each of the basis matrices $E_{ij}$ that is all zeros except that it is 1 at position $ij$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/105733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding the n-th root of a complex number I am trying to solve $z^6 = 1$ where $z\in\mathbb{C}$.
So What I have so far is :
$$z^6 = 1 \rightarrow r^6\operatorname{cis}(6\theta) = 1\operatorname{cis}(0 + \pi k)$$
$$r = 1,\ \theta = \frac{\pi k}{6}$$
$$k=0: z=\operatorname{cis}(0)=1$$
$$k=1: z=\operatorname{cis}\left(\fr... | I will solve $z^6=1$ where $z∈\mathbb{C}$. First I will take off all data: $r=1$, $\theta = 0$, $n=6$ and $k=0,1,2,3,4,5$ then
For $k=0: w_1=\sqrt[6]{1}[\cos{\frac{2\pi (0)}{6}}+ i\sin{\frac{2\pi (0)}{6}}]=1[\cos{0}+i \sin{0}]$ therefore $z_1=\cos{0}+i\sin{0}$ finally $z_1=1$
For $k=1: w_2=\sqrt[6]{1}[\cos{\frac{2\pi (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Solving Triangles (finding missing sides/angles given 3 sides/angles) What is a general procedure for "solving" a triangle—that is, for finding the unknown side lengths and angle measures given three side lengths and/or angle measures?
| First, some notation: let $A$, $B$, and $C$ be the measures of the three angles and let $a$, $b$, and $c$ be the lengths of the sides opposite those angles, respectively. Now, let's look case-by-case at the possible sets of information we could have.
SSS
Let's start with the case where we know the three sides, $a$, $b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 1,
"answer_id": 0
} |
Solving modular equations Is there a procedure to solve this or is it strictly by trial and error?
$5^x \equiv 5^y \pmod {39}$ where $y > x$.
Thanks.
| We can rewrite your problem as
$$5^{y-x}=1\bmod 39.$$
So we are trying to find the positive integers $k$ such that $5^k=1\bmod{39}$.
Since we can find the prime power factorization of $39$ easily, our problem reduces to finding the positive $n$ such that (simultaneously) $5^n=1\bmod 3$ and $5^n=1 \bmod{13}$.
If $p$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
The sum of the coefficients of $x^3$ in $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$ I know how to solve such questions when it's like $(x+y)^n$ but I'm not sure about this one:
In $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$, What's the sum of the
coefficients of $x^3$?
| The multinomial expansion gives $\mathbf{S}= (1 - \frac{x}{2} + \frac{1}{\sqrt{x}})^8 = \sum_{k_1+k_2 +k_3=8} \binom{8}{k_1,k_2,k_3} (1)^{k_1}(\frac{-x}{2})^{k_2} (\frac{1}{\sqrt{x}})^{k_3}.$ Simplifying we get $\mathbf{S}=\sum_{k_1+k_2 +k_3=8} \binom{8}{k_1,k_2,k_3} (\frac{-1}{2})^{k_2} (x)^{k_2 -k_3/2}.$ Now identify... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Showing $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$ Given that n is a positive integer show that $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$.
I'm thinking that I should be using the property of gcd that says if a and b are integers then gcd(a,b) = gcd(a+cb,b). So I can do things like decide that $\gcd(n^3 + 1, n^2 + 2) = ... | I'm carrying out a congruence procedure, so that you have different approaches.
If $p \, | \, n^3 + 1$ and $p \, | \, n^2 + 2$, then $2n \equiv 1 \pmod p$, which means
$$
-8 \equiv 8n^3 = (2n)^3 \equiv 1^3 \equiv 1 \pmod p,
$$
hence $9 \equiv 0 \pmod p$ and assuming $p$ is a prime means $p = 3$. This means the $\gc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 1
} |
Area of a trapezoid from given the two bases and diagonals Find the area of trapezoid with bases $7$ cm and $20$ cm and diagonals $13$ cm and $5\sqrt{10} $ cm.
My approach:
Assuming that the bases of the trapezoid are the parallel sides, the solution I can think of is a bit ugly,
*
*Find the other two non-parallel ... | First Solution: Let our trapezoid be $ABCD$ as in the diagram supplied by pedja. Let the diagonals meet at $O$.
Note that $\triangle OAB$ and $\triangle OCD$ are similar. Indeed we know the scaling factor. Since $AB=20$ and $CD=7$, the sides of $\triangle OCD$ are $\frac{7}{20}$ times the corresponding sides of $\tr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
What does $\sin^{2k}\theta+\cos^{2k}\theta=$?
What is the sum $\sin^{2k}\theta+\cos^{2k}\theta$ equal to?
Besides Mathematical Induction,more solutions are desired.
| I do not think there is a closed form for all values of $k$, but one can play around with trigonometric identities to simplify the expression for certain values of $k$. For instance:
*
*If $k=2$, then:
$$\sin^4 x + \cos^4 x = (1-\cos^2 x)^2 + \cos^4 x\\
= 1-2\cos^2x + 2\cos^4 x \\
= 1-2\cos^2x(1-\cos^2x)\\
= 1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/111265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Computing $ \sum_{n=1}^{\infty} \frac{\lfloor{\sqrt{n+1}}\rfloor-\lfloor{\sqrt{n}}\rfloor}{n}$ I would like to prove the existence and the exact value of the following series:
$$ \sum_{n=1}^{\infty} \frac{\lfloor{\sqrt{n+1}}\rfloor-\lfloor{\sqrt{n}}\rfloor}{n}$$
| If $ n+1 $ is a square, $\lfloor \sqrt{n+1} \rfloor=\sqrt{n+1} $
$$ 0<\sqrt{n+1}-\sqrt{n}=\frac{1}{\sqrt{n+1}+\sqrt{n}}<1$$
So: $$ \sqrt{n+1}-1<\sqrt{n}<\sqrt{n+1} $$
So: $$ \lfloor \sqrt{n} \rfloor= \sqrt{n+1}-1$$
$$ \lfloor{\sqrt{n+1}}\rfloor-\lfloor{\sqrt{n}}\rfloor=1 $$
I have proved:
$n+1$ is a square $\Longright... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Finding a simple expression for this series expansion without a piecewise definition I am doing some practice Calculus questions and I ran into the following problem which ended up having a reduction formula with a neat expansion that I was wondering how to express in terms of a series. Here it is: consider
$$
I_{n} =... | How about
$$\sum_{k=1}^{n} k \left(\frac{\pi}{2}\right)^{k-1} \cdot (-1)^{n+k+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/113655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Divide inside a Radical It has been so long since I have done division inside of radicals that I totally forget the "special rule" for for doing it. -_-
For example, say I wanted to divide the 4 out of this expression:
$\sqrt{1 - 4x^2}$
Is this the right way to go about it?
$\frac{16}{16} \cdot \sqrt{1 - 4x^2}$
$16 ... | Square roots obey the rule $\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}$ . You wanted to take the $4$ out of $\sqrt{1-4x^2}$. $1-4x^2=4\cdot\frac{1-4x^2}{4}$. So $\sqrt{1-4x^2}=\sqrt{4\cdot\frac{1-4x^2}{4}}=\sqrt{4}\cdot\sqrt{\frac{1-4x^2}{4}}=2\cdot\sqrt{\frac{1-4x^2}{4}}$. I think that's the best explanation I can give... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Solve $ x^2+4=y^d$ in integers with $d\ge 3$
Find all triples of integers $(x,y,d)$ with $d\ge 3$ such that $x^2+4=y^d$.
I did some advance in the problem with Gaussian integers but still can't finish it. The problem is similar to Catalan's conjecture.
NOTE: You can suppose that $d$ is a prime.
Source: My head
| Here's a proof sketch. It's missing two major steps Propositions 1 and 2, but I'll try to fill them in and come back with an update.
First consider $x$ and $y$ even. Then we can only have $x\equiv 4\pmod{16}$, $y\equiv 8\pmod{16}$ and hence that $d=3$, for which @André Nicolas showed that $x=y=2$ is the only solution.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/118941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 4
} |
finding variance from a piecewise function How do you calculate the variance of a piecewise function? For example, what would be the variance of the probability density function $f_x(x)= \frac{3}{4}, 0\leq x\leq 1; \frac{1}{4}, 2\leq x \leq 3; 0$ otherwise?
| Let $U$ be a continuous uniform random variable on a unit interval. It's mean and variance are
$$\mathbb{E}(U) = \frac{1}{2} \qquad \qquad
\mathbb{Var}(U) = \mathbb{E}(U^2) - \left(\mathbb{E}(U)\right)^2 = \frac{1}{12}
$$
Now, your continuous random variable $X$ is a component mixture of a uniform $U$ and shifted... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/119795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Compute $\lim \limits_{x\to\infty} (\frac{x-2}{x+2})^x$ Compute
$$\lim \limits_{x\to\infty} (\frac{x-2}{x+2})^x$$
I did
$$\lim_{x\to\infty} (\frac{x-2}{x+2})^x = \lim_{x\to\infty} \exp(x\cdot \ln(\frac{x-2}{x+2})) = \exp( \lim_{x\to\infty} x\cdot \ln(\frac{x-2}{x+2}))$$
But how do I continue? The hint is to use L Hop... | $$\lim_{x\to\infty} (\frac{x-2}{x+2})^x$$
$$\lim_{x\to\infty} (1-\frac{4}{x+2})^x = y$$
taking log on both sides we get
$$ln(y) = x ln (1- \frac{4}{x+2})$$
the expansion for $ln (1+r) $ is $ r- \frac{r^2}{2} +\frac{r^3}{3}$ ....
where r tends to zero
$$ln(y) = x ( \frac{-4}{x+2} - \frac{\frac{-4}{x+2}^2}{2} +\frac{\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/120006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
} |
Prove that the following integral is divergent $$\int_0^\infty \frac{7x^7}{1+x^7}$$
Im really not sure how to even start this. Does anyone care to explain how this can be done?
| The answer by Davide Giraudo has all the right elements, but I think the OP may appreciate to see all of the details spelled out. First of all, an improper integral is defined as a limit, as follows:
$$\int_0^\infty \frac{7x^7}{1+x^7} dx = \lim_{N\to \infty} \int_0^N \frac{7x^7}{1+x^7} dx.$$
Next, we use the fact that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/120090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Show that $\displaystyle{\frac{1}{9}(10^n+3 \cdot 4^n + 5)}$ is an integer for all $n \geq 1$ Show that $\displaystyle{\frac{1}{9}(10^n+3 \cdot 4^n + 5)}$ is an integer for all $n \geq 1$
Use proof by induction. I tried for $n=1$ and got $\frac{27}{9}=3$, but if I assume for $n$ and show it for $n+1$, I don't know what... | Any proof will need to make use of induction at some point.
*
*Note that $10^n \equiv 1 \bmod 9$ and $4^n \equiv (1,4,7) \bmod 9$. (You need induction to prove previous statements.) Hence, $3 \times 4^n \equiv 3 \bmod 9$. Assimilating these together, we get that $$\left(10^n + 3 \times 4^n + 5 \right) \equiv (1 + 3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/120649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 0
} |
Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$ My nephew asked me this, so I suggested him to sign up here. But anyways this question I was
trying to solve myself. I got part of the solution. Let me know the rest.
$(1)$
Solve the integral defined as
$\displaystyle{S_k = (-1)^k \int_0^1 (\log(\si... | As for the the problem (2) we have
$$
\begin{align}
S_k&=(-1)^k \int\limits_0^1\log^k(\sin\pi x)dx\\
&=\frac{(-1)^k}{\pi} \int\limits_0^\pi\log^k(\sin y)dy\\
&=\frac{(-1)^k}{\pi} \int\limits_0^{\pi/2}\log^k(\sin y)dy+
\frac{(-1)^k}{\pi} \int\limits_{\pi/2}^\pi\log^k(\sin y)dy\\
&=\frac{(-1)^k}{\pi} \int\limits_0^{\pi/2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 1
} |
Closed form for coefficient of $x^N$ of $\frac{1}{(1-x^2)(1-x^3)(1-x^5)}$ I'm trying to get a closed form for the generating function which counts the number of non-negative integer solutions to
$$2e_1+3e_2+5e_3 = N$$
I note that we have something like:
$$\frac{1}{(1-x^2)(1-x^3)(1-x^5)}$$
but can't get something to ... | According to Maple:
convert(1/(1-x^2)/(1-x^3)/(1-x^5),parfrac,x);
$$-1/30\, \left( x-1 \right) ^{-3}+1/8\, \left( x+1 \right) ^{-1}-{
\frac {77}{360}}\, \left( x-1 \right) ^{-1}+1/5\,{\frac {2+x+{x}^{2}+{
x}^{3}}{{x}^{4}+{x}^{3}+{x}^{2}+x+1}}+{\frac {7}{60}}\, \left( x-1
\right) ^{-2}+1/9\,{\frac {-x+1}{{x}^{2}+x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Why is the differentiability of a piecewise defined function studied using the definition of derivative? Why is the differentiability of a piecewise defined function often studied using the definition of derivative?
For example, let:
$$\begin{align*}
f(x)
&= x\cdot |x-1|\\
&= \left\{ \begin{array}{lcl}x (-x+1)& \te... | Consider the function
$$f(x) = \left\{\begin{array}{ll}
x^2\sin\left(\frac{1}{x}\right) &\text{if }x\neq 0,\\
0 &\text{if }x=0.
\end{array}\right.$$
The function is differentiable at $0$, with derivative equal to $0$:
$$\begin{align*}
\lim_{h\to 0}\frac{f(h)-f(0)}{h} &= \lim_{h\to 0}\frac{h^2\sin(\frac{1}{h})}{h} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to expand $\cos nx$ with $\cos x$? Multiple Angle Identities:
How to expand $\cos nx$ with $\cos x$, such as
$$\cos10x=512(\cos x)^{10}-1280(\cos x)^8+1120(\cos x)^6-400(\cos x)^4+50(\cos x)^2-1$$
See a list of trigonometric identities in english/ chinese
| You can always repeatedly use
$$\begin{align*}
\cos(a\pm b) &= \cos a\cos b \mp \sin a\sin b\\
\sin(a\pm b) &= \sin a\cos b \pm \cos a\sin b\\
\sin^2(r) &= 1-\cos^2(r).
\end{align*}$$
For example,
$$\begin{align*}
\cos(4x) &= \cos(2x+2x)\\
&= \cos(2x)^2 - \sin^2(2x)\\
&= \cos(2x)^2 - (1-\cos^2(2x))\\
&= 2\cos(2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 5
} |
The $n^{th}$ root of the geometric mean of binomial coefficients. $\{{C_k^n}\}_{k=0}^n$ are binomial coefficients. $G_n$ is their geometrical mean.
Prove
$$\lim\limits_{n\to\infty}{G_n}^{1/n}=\sqrt{e}$$
| $G_n$ is the geometric mean of $n+1$ numbers:
$$
G_n=\left[\prod_{k=0}^n{n\choose k}\right]^{\frac1{n+1}}
$$
or with $\log$ representing the natural logarithm (to the base $e$),
$$
\log G_n
= \frac1{n+1} \sum_{k=0}^n \log {n\choose k}
= \log n! - \frac2{n+1} \sum_{k=0}^n \log k!
\,.
$$
Stirling's approximation i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 2
} |
Evaluation of $ \lim_{ x \to \infty} x^{\frac{3}{2}}(\sqrt{x+2}-2\sqrt{x+1}+\sqrt{x})$ I am trying to calculate $$ \lim_{ x \to \infty} x^{\frac{3}{2}}(\sqrt{x+2}-2\sqrt{x+1}+\sqrt{x})$$
Whenever I try it all I can seem to reach is the conclusion that it goes to infinity, however when I try it in wolframalpha it gives... | $$\displaystyle \lim_{x \to \infty}x^{3/2}(\sqrt {x+2}+\sqrt x-2\sqrt{x+1})=\displaystyle \lim_{x \to \infty} \frac{x^2(-2x-2+2\sqrt{x^2+2x})}{\sqrt x(\sqrt {x+2}+\sqrt x+2\sqrt{x+1})}=$$
$$=\displaystyle \lim_{x \to \infty}\frac{2x^2(\sqrt{x^2+2x}-(x+1))(\sqrt{x^2+2x}+(x+1))}{\sqrt x (\sqrt {x+2}+\sqrt x+2\sqrt{x+1})(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
How to evaluate the integral: $\int_{0}^{1}\frac{dx}{1+x^{3}}$ I am a bit lost on how to evaluate the integral: $$\int_{0}^{1}\frac{dx}{1+x^{3}}$$
I tried the substitution: $y=x^{3}$, but I got a more complicated integrand. Any ideas?
| There's a factorization of a sum of two cubes:
$$
1+x^3 = (1+x)(1-x+x^2)
$$
This quadratic polynomial is irreducible unless you allow imaginary numbers since if you write it as $ax^2+bx+c$, it turns out that $b^2-4ac<0$.
So
$$
\frac{1}{1+x^3} = \frac{1}{(1+x)(1-x+x^2)} = \frac{A}{1+x} + \frac{Bx+C}{1-x+x^2}
$$
As usu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is this proof that if $a_{n+1} = \sqrt{2 + \sqrt{a_n}}$ and $a_1 = \sqrt{2}$, then $\sqrt{2} \leq a_n \leq 2$ correct? Let $a_1 = \sqrt{2}$ and $a_{n+1} = \sqrt{2 + \sqrt{a_n}}$. Now I want to show by induction that $\sqrt{2} \leq a_n \leq 2$ for all $n$.
The base case is $n=1$ and it is clear $\sqrt{2} \leq a_1 \leq ... | Note that you want to show the case $n$ implies $n+1$, but you're going the other way.
The good thing about this sequence is the inductive step is indeed easy.
The base case is $\sqrt 2 \leq a_1 \leq 2$
We now assume $\sqrt 2 \leq a_n \leq 2$
Take a square root, then add two, then take a square root.
$$\sqrt{\root 4 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/138275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Showing a series is a solution to a differential equation I am attempting to show that the series $y(x)\sum_{n=0}^{\infty} a_{n}x^n$ is a solution to the differential equation $(1-x)^2y''-2y=0$ provided that $(n+2)a_{n+2}-2na_{n+1}+(n-2)a_n=0$
So i have:
$$y=\sum_{n=0}^{\infty} a_{n}x^n$$
$$y'=\sum_{n=0}^{\infty}na_{n}... | Everything seems correct. Before you expand out powers of $n$, notice that equating like powers of $x^n$ in your last sum gives:
$(n+2)(n+1)a_{n+2}-2n(n+1)a_{n+1}+[n(n-1)-2]a_n=0$
Notice that $[n(n-1)-2]=n^2-n-2=(n+1)(n-2)$, so since $n\geq 0$, you can divide the recurrence equation by $(n+1)$ to get the desired result... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/138520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Determinant of a 3x3 matrix with 6 unknowns given the determinants of two 3x3 matrices with same unknowns? Given:
$$
det(A) = 3 \\ det(B) = -4
$$
$$
A =
\begin{pmatrix}
a & b & c \\
1 & 1 & 1\\
d & e & f
\end{pmatrix} \\
B =
\begin{pmatrix}
a & b & c \\
1 & 2 & 3 \\
d & e & f
\end{pmatrix} \\
C =
\begin{pmatrix}
a & b ... | The determinant is a multilinear function of the rows (or columns). Since $(4,6,8) = 2(1,1,1)+2(1,2,3)$, we have $\det C = 2 \det A + 2 \det B$. Hence the answer is $-2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/140144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How can I show using mathematical induction that $\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n} = \frac{2^n - 1}{2^n}$ How can I show using mathematical induction that $\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n} = \frac{2^n - 1}{2^n}$
Edit: I'm specifically stuck on showing that $\frac{2^n - 1}{2^n} + \fra... | For what is worth:
$$S_n = \frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{{{2^n}}}$$
Then
$$\frac{{{S_n}}}{2} = {S_n} + \frac{1}{{{2^{n + 1}}}} - \frac{1}{2} \Rightarrow {S_n} = 2{S_n} + \frac{1}{{{2^n}}} - 1 \Rightarrow 1 - \frac{1}{{{2^n}}} = {S_n}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/141126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
laplace form and matrix exponential Given the matrix $(I-A)^{-1}$ and $B$, can we compute $e^{A+B}$, where $e^X$ is defined to be $\sum_{i=0}^{\infty} \frac{X^i}{i!}$.
(Note that $A$ and $B$ do not commute, and hence $e^A \cdot e^B \neq e^{A+B}$).
Now I've observed that Laplace transformation might be a useful tool. I'... | $e^{A+B}$ is not uniquely determined by $e^A$ and $e^B$.
First take $A = \left[ \begin{array}{cc} 0 & -\pi \\\ \pi & 0 \end{array} \right]$ and $B = \left[ \begin{array}{cc} \pi & 0 \\\ 0 & -\pi \end{array} \right]$. Then $A + B$ squares to zero, so we have
$$e^A = \left[ \begin{array}{cc} \cos \pi & - \sin \pi \\\ \si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/143907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the $x(t)$ function of $\dot{v} = a v² + bv + c$ to obtain $x(t)$ How to solve $$\frac{dv}{dt} = av^2 + bv + c$$ to obtain $x(t)$, where $a$, $b$ and $c$ are constants, $v$ is velocity, $t$ is time and $x$ is position. Boundaries for the first integral are $v_0$, $v_t$ and $0$, $t$ and boundaries for the second... | As suggested by Arturo:
$$ \begin{eqnarray}
\int_{0}^{t} \frac{\dot{v}(t)}{a v(t)^2+ b v(t) + c} \mathrm{d} t &=& \int_0^t \mathrm{d} t \\
\int_{v_0}^{v_t} \frac{\mathrm{d} u}{a u^2 + b u + c} &=& t \\
\int_{v_0}^{v_t} \frac{4 a }{\left(2 a u + b\right)^2 + 4 a c - b^2} \mathrm{d} u &=& t \\
\left.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/145570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Number of partitions contained within Young shape $\lambda$ It is well known that the number of partitions contained within an $m\times n$ rectangle is $\binom{m+n}{n}$.
Furthermore, it is not difficult to calculate the number of partitions contained within a Young shape $\lambda$, where $\lambda $ is also a partition,... | Ira Gessel has very kindly explained how this can be solved by counting nonintersecting lattice paths. Again, working through the example of $(3,2,1,1)$ will explain the general approach.
First, convert the problem to counting distinct part partitions contained in $(3+3,2+2,1+1,1+0) = (6,4,2,1)$. Now a collection o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
$3^2 \ 5^2 \ldots (p-2)^2 \equiv (-1)^{\frac{p+1}{2}} \ (\mathrm{mod} \ p)$
Possible Duplicate:
Why is the square of all odds less than an odd prime $p$ congruent to $(-1)^{(p+1)/(2)}\pmod p$?
Why is $3^2 \ 5^2 \ldots (p-2)^2 \equiv (-1)^{\frac{p+1}{2}} \ (\mathrm{mod} \ p)$, where $p$ is an odd prime?
I can't seem... | Since $a\equiv -(p-a)\pmod{p}$, we can write
\begin{align}
3^2 \cdot 5^2 \cdots (p-2)^2&\equiv \left(3\cdot 5\cdots (p-2)\right)\times\left((p-3)\cdot (p-5)\cdots 2\right)\times (-1)^{(p-3)/2}\\
&\equiv (-1)^{(p-3)/2}(2\cdot 3\cdots (p-2))\cdot (p-1)\cdot (p-1)\\
&\equiv (-1)^{(p-1)/2}(p-1)!\\
&\equiv (-1)^{(p+1)/2}\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/147438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$. In this (btw, nice) answer to Twin primes of form $2^n+3$ and $2^n+5$, it was said that:
If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$?
I'm not familiar with these kind of calculations, so I'd like to see, if my answer is correct:
*
*Let $n=3k+2$ so then
$2^{3k+2}+3\equiv 2^{... | It doesn't require factorization or any other such ingenuity, just simple modular arithmetic:
$$\rm mod\ 7\!:\ \ 3 + 2^{\:\!2+3\:\!K} = 3 + 4\cdot 8^K \equiv 3 + 4\cdot 1^K\equiv 3+4 \equiv 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/148418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Evaluating $\lim\limits_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$ Evaluate $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$
| another method, using Stolz–Cesàro theorem:
let ${ x }_{ n }=\left( p+1 \right) \left( { 1 }^{ p }+{ 2 }^{ p }+...+{ n }^{ p } \right) -{ n }^{ p+1 },{ y }_{ n }=\left( p+1 \right) { n }^{ p }$
$$\lim _{ x\rightarrow \infty }{ \frac { { x }_{ n+1 }-{ x }_{ n } }{ { y }_{ n+1 }-{ y }_{ n } } = } \lim _{ x\rightarrow \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/149142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 7,
"answer_id": 0
} |
Long division in integration by partial fractions I am trying to figure out what my book did, I can't make sense of the example.
"Since the degree of the numberator is greater than the degree of the denominator, we first perform the long division. This enables us to write
$$\int \frac{x^3 + x}{x -1} dx = \int \left(x^2... | Your first term is indeed $x^2$. Then $$x^3+x-x^2(x-1)=x^3+x-(x^3-x^2)=x^2+x,$$ and so your next term is $x$. Then $$x^2+x-x(x-1)=x^2+x-(x^2-x)=2x,$$ so your next term is $2$. Finally, $2x-2(x-1)=2$, so your remainder is $2$. I suspect you simply made an arithmetic error. It always helps to check your answer, and indee... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Sum with binomial coefficients: $\sum_{k=1}^m \frac{1}{k}{m \choose k} $ I got this sum, in some work related to another question:
$$S_m=\sum_{k=1}^m \frac{1}{k}{m \choose k} $$
Are there any known results about this (bounds, asymptotics)?
| We will make use of Euler-Maclaurin to get the asymptotic. The final asymptotic is $$\sum_{k=1}^{m} \dfrac1k\dbinom{m}{k} = \dfrac{2^{m+1}}{m} \left(\sum_{n=0}^{N} \dfrac{2^n \Gamma(n+1/2)}{m^n \sqrt{\pi}} \right) + \mathcal{O} \left( \dfrac{2^{m+1}}{m^{N+2}}\right)$$
Let us denote $\displaystyle \sum_{k=1}^{m} \dfrac1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 7,
"answer_id": 0
} |
Finite Sum of Power? Can someone tell me how to get a closed form for
$$\sum_{k=1}^n k^p$$
For $p = 1$, it's just the classic $\frac{n(n+1)}2$.
What is it for $p > 1$?
| By Binomial Series
$(n+1)^x=1 + {x \choose 1}\sum_{k=1}^n{k^{x-1}} + {x \choose 2}\sum_{k=1}^n{k^{x-2}}+{x \choose 3}\sum_{k=1}^n{{k^{x-3}}}
...+{x \choose x-1}\sum_{k=1}^n{{k^{x-x+1}}}+{x \choose x}\sum_{k=1}^n{{k^{x-x}}}$
which becomes
$(n+1)^x = 1 + {x \choose 1}\sum_{k=1}^n{k^{x-1}} + {x \choose 2}\sum_{k=1}^n{k^{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 1
} |
Simplify these expressions with radical sign 2 My question is
1) Rationalize the denominator:
$$\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$$
My answer is:
$$\frac{\sqrt{12}+\sqrt{18}-\sqrt{30}}{18}$$
My question is
2) $$\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{2}-\sqrt{3}-\sqrt{5}}$$
My answer is: $$\frac... | \begin{align}
\dfrac1{\sqrt{2} + \sqrt{3} + \sqrt{5}} & = \dfrac{\sqrt{2} + \sqrt3 - \sqrt5}{(\sqrt{2} + \sqrt{3} + \sqrt{5})(\sqrt{2} + \sqrt{3} - \sqrt{5})}\\
& = \dfrac{\sqrt{2} + \sqrt3 - \sqrt5}{(\sqrt{2} + \sqrt{3})^2 - 5}\\
& = \dfrac{\sqrt{2} + \sqrt3 - \sqrt5}{(2+3+2 \sqrt{6}) - 5}\\
& = \dfrac{\sqrt{2} + \sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find the square root of the polynomial My question is:
Find the square root of the polynomial-
$$\frac{x^2}{y^2} + \frac{y^2}{x^2} - 2\left(\frac{x}y + \frac{y}x\right) + 3$$
| The polynomial is
$ \frac{x^2}{y^2} + \frac{y^2}{x^2} - 2\left( \frac{x}{y} + \frac{y}{x} \right) + 3$
Denote $ z = \frac{x}{y}$ and have
$ z^2 + z^{-2} - 2(z+z^{-1}) + 3 $
Assume $y \ne 0$ and multiply by $z^2$ and have
$ z^4 + 1 - 2z^3 - 2z + 3z^2 = 0$
Instructions how to solve a general quartic equation can be found... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
determine whether this series converges for this value of z Does $$f(z)=\displaystyle \sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3}z^n$$ converge for $z=\frac{-3}{2}$?
| $$ \frac{2^n + n^2}{3^n + n^3} z^n = \frac{2^n}{3^n + n^3} z^n + \frac{n^2}{3^n + n^3} z^n $$
$$ \frac{n^2}{3^n + n^3} z^n < n^2 \left ( \frac{z}{3} \right )^n = \frac{n^2 }{(-2)^n } \text{ Which converges from Ratio test }$$
$$ \frac{2^n}{3^n + n^3} z^n = \left ( \frac{2}{3} z\right )^n \frac{1}{1 + \frac{n^3}{3^n}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluation of $\lim\limits_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$ One of the previous posts made me think of the following question: Is it possible to evaluate this limit without L'Hopital and Taylor?
$$\lim_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$$
| Encouraged by Hans Lundmark's answer, I'm posting my own solution without derivatives and integrals.
The triple-angle formula for $\tan$ is $$\tan 3\theta = \frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}.$$
Suppose $\lim_{x\to0}(\tan x-x)/x^3 = c$. Letting $x = 3\theta$, we then have
$$\begin{align}
c &= \lim_{x\to0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 1
} |
Evaluting: $\int\frac{1}{(1+\tan x)^2} dx$ I can solve this integral
$$
\int\frac{1}{(1+\tan x)^2} dx
$$
using the substitution $t=\tan x$ i.e $x=\arctan t$. Does anyone know another way to solve this integral?
| Using J.M.'s suggestion
$$ \int \frac{1}{( 1 + \tan x)^2} dx = \int \frac{\frac 12 (1 + \cos 2x)}{ \sin ^2x +2 \sin x \cos 2 + \cos ^2 x} dx = \frac 12 \int \frac{1 + \cos 2x}{1 + \sin 2x}dx$$
$$ = \frac 12 \left [ \int \frac 1 {1 + \sin 2x} dx + \int \frac{\cos 2x}{1 + \sin 2x}dx\right ]$$
$$ = \frac 1 2 \frac{\sin x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
What is the result of sum $\sum\limits_{i=0}^n 2^i$
Possible Duplicate:
the sum of powers of $2$ between $2^0$ and $2^n$
What is the result of
$$2^0 + 2^1 + 2^2 + \cdots + 2^{n-1} + 2^n\ ?$$
Is there a formula on this? and how to prove the formula?
(It is actually to compute the time complexity of a Fibonacci recur... | How much is a direct summation worth?
$$\begin{align*}
1 + \sum_{i=0}^n 2^i &= 1 + (2^0 + 2^1 + 2^2 + \cdots + 2^n)\\
&= (2^0 + 2^0) + (2^1 + 2^2 + \cdots + 2^n)\\
&= 2^1 + (2^1 + 2^2 + \cdots + 2^n)\\
&= (2^1 + 2^1) + (2^2 + \cdots + 2^n)\\
&= 2^2 + (2^2 + \cdots + 2^n)\\
\vdots &= \ddots\\
&= 2^n + (2^n)\\
&= 2^{n+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 2
} |
Question on the unsolvability a group Let $G$ be a finite group. Let $\pi(G)=\{2,3,5\}$ be the set of prime
divisors of its order. If 6 divide the number of Sylow 5-subgroups of G
and 10 divide the number of Sylow $3$-subgroups of $G$, then whether the
group $G$ group with those properties is unsolvable?
In particular ... | No. There is a solvable group of order $2^{22} 3^5 5^3$ with $n_2=1$, $n_3=2^{18} 5^2$, and $n_5 = 2^{20} 3^4$.$\newcommand{\GF}{\operatorname{GF}}\newcommand{\AGL}{\operatorname{AGL}}$
$$G =
\left(\left(3\ltimes\GF(5^2)\right) \ltimes \left(\GF(2^4)^3\right)\right) ~ \times ~ \left(\left(5\ltimes\GF(3^4)\right) \ltim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A simple quadratic inequality For positive integers $n\ge c\ge 5$, why does
$$c+2(n-c)+\frac{(n-c)^2}{4}\le\frac{(n-1)^2}{4}+1\text{ ?}$$
| If you expand the two sides, you will get
\begin{equation}
\frac{c^2}{4}-\frac{cn}{2}-c+\frac{n^2}{4}+2n\leq\frac{n^2}{4}-\frac{n}{2}+\frac{5}{4}
\end{equation}
\begin{equation}
\frac{c^2}{4}-\frac{5}{4}-c\leq n\frac{c-5}{2}
\end{equation}
If c=5 than the inequality is satisfied, assuming $5\leq c$ we have
\begin{equat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Compute: $\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$ I try to solve the following sum:
$$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$$
I'm very curious about the possible approaching ways that lead us to solve it. I'm not experienced with these sums, and any hint, suggestion is v... | Let's write:
$$f(x) = \sum_{k = 1}^{+\infty}\sum_{n = 1}^{+\infty} \frac{x^{n+k+2}}{nk (n+k+2)}$$
then:
$$f'(x) = \sum_{k = 1}^{+\infty}\sum_{n = 1}^{+\infty} \frac{x^{n+k+1}}{nk} = - \sum_{k = 1}^{+\infty}\frac{x^{k+1} \ln (1-x)}{k}$$
We want to know the value of $f(1)$, so we integrate:
$$f(1) = - \sum_{k = 1}^{+\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Tangent line of parametric curve I have not seen a problem like this so I have no idea what to do.
Find an equation of the tangent to the curve at the given point by two methods, without elimiating parameter and with.
$$x = 1 + \ln t,\;\; y = t^2 + 2;\;\; (1, 3)$$
I know that $$\dfrac{dy}{dx} = \dfrac{\; 2t\; }{\dfrac{... | *
*Eliminating the parameter $t$. The given system of two parametric equations $$\begin{eqnarray*}\left\{
\begin{array}{c}
x=1+\ln t \\
y=t^{2}+2
\end{array}
\right. \end{eqnarray*} \tag{A}$$ is equivalent successively to
$$\begin{eqnarray*}
\left\{
\begin{array}{c}
x-1=\ln t \\
y=t^{2}+2
\end{array}
\right. \Left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Triples of positive real numbers $(a,b,c)$ such that $\lfloor a\rfloor bc=3,\; a\lfloor b\rfloor c=4,\;ab\lfloor c\rfloor=5$ Find the all ordered triplets of positive real numbers $(a,b,c)$ such that: $$\lfloor a\rfloor bc=3,\quad a\lfloor b\rfloor c=4,\quad ab\lfloor c\rfloor=5,$$
where $\lfloor x\rfloor$ is the grea... | Now we have by dividing equations and putting together the ratios we get the ratio $$\frac a{\lfloor a\rfloor}:\frac b{\lfloor b\rfloor}:\frac c{\lfloor c\rfloor}=20:15:12$$
This shows that $\frac a{\lfloor a\rfloor}\ge\frac53$ since $\frac c{\lfloor c\rfloor}\ge1$. It is checkable that $\frac a{\lfloor a\rfloor}\ge\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
How can I calculate this limit: $\lim\limits_{x\rightarrow 2}\frac{2-\sqrt{2+x}}{2^{\frac{1}{3}}-(4-x)^\frac{1}{3}}$? I was given this limit to solve, without using L'Hospital rule. It's killing me !! Can I have the solution please ?
$$\lim_{x\rightarrow 2}\frac{2-\sqrt{2+x}}{2^{\frac{1}{3}}-(4-x)^\frac{1}{3}}$$
| $$ \lim_{x\rightarrow 2}\frac{2-\sqrt{2+x}}{2^{\frac{1}{3}}-(4-x)^\frac{1}{3}} $$
$$ = \lim_{x\rightarrow 2} \frac{2-\sqrt{2+x}}{2^{\frac{1}{3}}-(4-x)^\frac{1}{3}} \times \frac{2+\sqrt{2+x}}{2+\sqrt{2+x}} \times \frac{2^{2/3} +2^{1/3}(4-x)^{1/3} +(4-x)^{2/3} }{2^{2/3} +2^{1/3}(4-x)^{1/3} +(4-x)^{2/3}}$$
$$ = \lim_{x\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Linear algebra: power of diagonal matrix? Let A = $\begin{pmatrix} 3 & -5 \\ 1 & -3 \end{pmatrix}$. Compute $A^{9}$. (Hint: Find a matrix P such that $P^{-1}AP$ is a diagonal matrix D and show that $A^{9}$= $PD^{9}P^{-1}$
Answer: $\begin{pmatrix} 768 & -1280 \\ 256 & -768 \end{pmatrix}$
I keep getting $\begin{pmatrix} ... | With $P=\begin{pmatrix} 5&1 \\1 & 1 \end{pmatrix}$ we have $P^{-1}AP = \left(\begin{array}{rr}2 & 0\\0 & -2\end{array}\right)$ then you can use your formula $PA^{9}P^{-1}$ and calculate $PA^{9}P^{-1}$.
$ \left(\begin{array}{rr}2 & 0\\0 & -2\end{array}\right)^9=\left(\begin{array}{rr}\,\,2^9 & 0\,\,\,\,\,\,\,\\0 & \,(-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Calculate $\lim\limits_{ x\rightarrow 100 }{ \frac { 10-\sqrt { x } }{ x+5 } }$ $$\lim_{ x\rightarrow 100 }{ \frac { 10-\sqrt { x } }{ x+5 } } $$
Could you explain how to do this without using a calculator and using basic rules of finding limits?
Thanks
| Recall that $\lim_{x \to a} f(x) = f(a)$ if $f(x)$ is continuous at $a$. The function $f(x) = \dfrac{10-\sqrt{x}}{x+5}$ is continuous at $100$ since the numerator $10-\sqrt{x}$ is continuous for all $x > 0$ and the denominator $5+x$ is continuous for all $x$. Hence, the function $f(x) = \dfrac{10-\sqrt{x}}{x+5}$ is con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Power (Laurent) Series of $\coth(x)$ I need some help to prove that the power series of $\coth x$ is:
$$\frac{1}{x} + \frac{x}{3} - \frac{x^3}{45} + O(x^5) \ \ \ \ \ $$
I don't know how to derive this, should I divide the expansion of $\cosh(x)$ by the expansion of $\sinh(x)$? (I've tried but without good results :( )
... | Long division of series with $\cosh(x) = 1 + \dfrac{x^2}{2} + \dfrac{x^4}{24} + \ldots$
and $\sinh(x) = x + \dfrac{x^3}{6} + \dfrac{x^5}{120} + \ldots$. Unfortunately I don't know how to typeset this nicely in LaTeX.
First term is $1/x$,
$$1 + \dfrac{x^2}{2} + \dfrac{x^4}{24} + \ldots - \dfrac{1}{x} \left(x + \dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/167149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Given that $x=\dfrac 1y$, show that $∫\frac {dx}{x \sqrt{(x^2-1)}} = -∫\frac {dy}{\sqrt{1-y^2}}$ Given that $x=\dfrac 1y$, show that $\displaystyle \int \frac 1{x\sqrt{x^2-1}}\,dx = -\int \frac 1{\sqrt{1-y^2}}\,dy$
Have no idea how to prove it.
here is a link to wolframalpha showing how to integrate the left side.
| Given that $x=1/y$
Put $x = \dfrac{1}{y},dx = \dfrac{-1}{y^2}.dy$:
$$\int \dfrac{1}{(x\sqrt{(x^2-1)})}dx$$
$$=\int \dfrac{1}{(\frac{1}{y} \sqrt{((\frac{1}{y})^2-1)})}\cdot\dfrac{-1}{y^2}dy$$
$$\int \dfrac{1}{\frac{1}{y} \sqrt{\frac{1-y^2}{y^2}}}\cdot \dfrac{-1}{y^2}dy$$
Simplify and you get your answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/168195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
A problem dealing with even perfect numbers. Question: Show that all even perfect numbers end in 6 or 8.
This is what I have. All even perfect numbers are of the form $n=2^{p-1}(2^p -1)$ where $p$ is prime and so is $(2^p -1)$.
What I did was set $2^{p-1}(2^p -1)\equiv x\pmod {10}$ and proceeded to show that $x=6$ or $... | In this proof, n is any integer(not just primes)
The number 2^(n-1) last digit repeats in the four number cycle(2,4,8,6)
The number 2^(n) -1 last digit repeats in the four number cycle (3,7,5,1)
The product's last digit repeats in the four number cycle (6,8,0,6)
If n is odd, the pattern repeats (8,6)
You will fin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
how to solve system of linear equations of XOR operation? how can i solve this set of equations ? to get values of $x,y,z,w$ ?
$$\begin{aligned} 1=x \oplus y \oplus z \end{aligned}$$
$$\begin{aligned}1=x \oplus y \oplus w \end{aligned}$$
$$\begin{aligned}0=x \oplus w \oplus z \end{aligned}$$
$$\begin{aligned}1=w \oplus... | As others have noted, all the usual methods of solving systems of linear equations (such as Gaussian elimination) in the field of real numbers work just as well in the finite field of integers modulo 2, also known as $GF(2)$.
In this field, addition corresponds to the XOR operation, while multiplication corresponds to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 0
} |
Find all real solutions to $8x^3+27=0$
Find all real solutions to $8x^3+27=0$
$(a-b)^3=a^3-b^3=(a-b)(a^2+ab+b^2)$
$$(2x)^3-(-3)^3$$ $$(2x-(-3))\cdot ((2x)^2+(2x(-3))+(-3)^2)$$ $$(2x+3)(4x^2-6x+9)$$
Now, to find solutions you must set each part $=0$. The first set of parenthesis is easy $$(2x+3)=0 ; x=-\left(\frac{3}{... | We know that $-3/2$ is a solution. Then we divide $x^3+27$ by $x+3/2$. Hence we have
$x^3+27=(x+3/2)(8x^2-12x+18)=0$. But, 8x^2-12x+18 don't have real solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/171682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$
Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$
$$\tan x+\sec x=2\cos x$$
$$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x$$
$$\left(\dfrac{\sin x+1}{\cos x}\right)=2\cos x$$
$$\sin x+1=2\cos^2x$$
$$2\cos^2x-\sin x... | Of course, $x$ cannot be an integer multiple of $\pi$ plus $\frac{\pi}{2}$. We have here a quadratic equation after a little manipulation, which you, yourself, started.
$$\frac{\sin x}{\cos x}+\frac{1}{\cos x}=2\cos x$$
After multiplying by $\cos x$, using the identity $1=\sin^2 x+\cos^2 x$, and bringing everything to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
show that $x^2+y^2=z^5+z$ Has infinitely many relatively prime integral solutions How to show that this equation:
$$x^2+y^2=z^5+z$$
Has infinitely many relatively prime integral solutions
| It is necessary to write down the formula!
In the equation:
$X^2+Y^2=Z^5+Z$
I think this formula should be written in a more general form:
$Z=a^2+b^2$
$X=a(a^2+b^2)^2+b$
$Y=b(a^2+b^2)^2-a$
And yet another formula:
$Z=\frac{a^2+b^2}{2}$
$X=\frac{(a-b)(a^2+b^2)^2-4(a+b)}{8}$
$Y=\frac{(a+b)(a^2+b^2)^2+4(a-b)}{8}$
$a,b$ - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The number $\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$ is always an integer For each $n$ consider the expression $$\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$$
I am trying to prove by induction t... | Try writing
$$
\left(\frac{1+\sqrt{5}}{2}\right)^n = \frac{a_n+b_n\sqrt{5}}{2}
$$
with $a_1=b_1=1$ and
$$
\frac{a_{n+1}+b_{n+1}\sqrt{5}}{2} = \left(\frac{1+\sqrt{5}}{2}\right)\left(\frac{a_n+b_n\sqrt{5}}{2}\right)
$$
see what you get, then repeat as much as you need to with $\left(\frac{1-\sqrt{5}}{2}\right)^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/174438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proving Quadratic Formula purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become $\dfrac{b^2}{4a^2}$?
| \begin{align*}
ax^2+bx+c &= 0 \\
ax^2+bx &= -c \\
x^2+\frac{b}{a}x &= -\frac{c}{a} \\
x^2+2x\Bigl(\frac{b}{2a}\Bigr)+\Bigl(\frac{b}{2a}\Bigr)^2-\Bigl(\frac{b}{2a}\Bigr)^2 &= -\frac{c}{a} \\
\Bigl(x+\frac{b}{2a}\Bigr)^2-\frac{b^2}{4a^2} &= -\frac{c}{a} \\
\Bigl(x+\frac{b}{2a}\Bigr)^2 &= \frac{b^2-4ac}{4a^2} \\
x+\frac{b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 6
} |
Prove $\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots$ converges to $\frac 1 2 $ Show that
$$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \cdots = \frac{1}{2}.$$
I'm not exactly sure what to do here, it seems awfully similar to Zeno's paradox.
If the series continues infin... | You can prove it with partial sums:
$$
S_n=\sum_{k=1}^n\frac{1}{(2k-1)(2k+1)}=\sum_{k=1}^n\left(\frac{1}{2(2k-1)}-\frac{1}{2(2k+1)}\right)=\frac{1}{2}\left(\sum_{k=1}^n\frac{1}{2k-1}-\sum_{k=2}^{n+1}\frac{1}{2k-1}\right)
$$ $$
=\frac{1}{2}\left(\frac{1}{2(1)-1}-\frac{1}{2(n+1)-1}\right)=\frac{1}{2}-\frac{1}{2(2n+1)}
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 0
} |
Prove that for any $x \in \mathbb N$ such that $xProve that every positive integer $x$ with $x<n!$ is the sum of at most $n$ distinct divisors of $n!$.
| People do not seem to be going along with my comment. So this is CW, and directly from the answer by Sanchez.
For $n=2,$ we need only 1 divisor of $2!,$ as $1=1.$
For $n=3,$ we need only 2 divisors of $3!=6,$ as $1=1, 2=2,3=3,4=3+1,5=3+2.$
Induction hypothesis: for some $n \geq 2,$ we need at most $(n-1)$ distinct divi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
} |
Image of $\partial B_r $ under an holomorphic function Let
\begin{equation*}
\begin{split}
f \colon & \mathbb C \setminus \left\{0\right\} \to \mathbb C \\
& z \mapsto \frac{1}{2}\left( z+ \frac{1}{z}\right)
\end{split}
\end{equation*}
I am asked to find the image of $\partial B_r := \{z \in \mathbb C : \vert z \vert... | We have, writing $a+ib:=re^{it}$ that
\begin{align}
f(re^{it})&=\frac 12\left(a+ib+\frac 1{a+ib}\frac{a-ib}{a-ib}\right)\\
&=\frac 12\left(a+ib+\frac{a-ib}{r^2}\right)\\
&=\frac 12\left(a\left(1+\frac 1{r^2}\right)+ib\left(1-\frac 1{r^2}\right)\right)\\
&=\frac 12\left(r\cos \theta\frac{r^2+1}{r^2}+ir\sin\theta\frac{r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/178414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Two sums with Fibonacci numbers
*
*Find closed form formula for sum: $\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{n} \frac{F_{2k}F_{n-k}}{10^n}$
*Find closed form formula for sum: $\displaystyle\sum_{k=0}^{n}\frac{F_k}{2^k}$ and its limit with $n\to +\infty$.
First association with both problems: generating... | The first one can be solved using the fact that the generating
function of the Fibonacci numbers is
$$\frac{z}{1-z-z^2}.$$
Introduce the function
$$f(z) = \sum_{n\ge 0} z^n \sum_{k=0}^n \frac{F_{2k} F_{n-k}}{10^n}$$
so that we are interested in $f(1).$
Re-write $f(z)$ as follows:
$$f(z) = \sum_{k\ge 0} F_{2k}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/179855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Prove $\cos^2x\sin^4x = \frac{1}{32}(2-\cos(2x)-2\cos(4x)+\cos(6x))$ I need help with a problem some may consider odd but here it is.
I have the following trig identity I been working on and I managed to get it to.
$$\cos^2x\sin^4x=\frac{3}{16}-\frac{\cos(2x)}{4}+\frac{3\cos(2x)}{16}+\frac{1}{8}(1+\cos(4x))+\frac{1}{32... | The equation you have written is not correct in general.
What you can do is note that there exists some $\theta \in [0, \pi)$ such that $\sin^2 x = \sin \theta$.
Then $1 - \sin^2 x = 1 - \sin \theta \Rightarrow \cos^2 x = 1 - \sin \theta$.
Thus we know that $\cos^2 x \sin^4 x = (1 - \sin \theta) \sin^2 \theta$.
Let $1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Computing sign table for $2^k$ factorial experiment design all!
I need to compute the sign table for a generic $2^k$ factorial design. For $k$ factors we compute $2^k$ experiments and need to compute a $2^k \times 2^k$ matrix, as the following example for $k=3$:
\begin{matrix}
& I & A & B & C & AB & AC & BC & ABC\\
1 ... | It's hard to see the bit structure in the table the way you've ordered it. If you write it like this:
$$
\begin{matrix}
& I & A & B & AB & C & AC & BC & ABC\\
8 & +1 & +1 & +1 & +1 & +1 & +1 & +1 & +1\\
7 & +1 & -1 & +1 & -1 & +1 & -1 & +1 & -1\\
6 & +1 & +1 & -1 & -1 & +1 & +1 & -1 & -1\\
5 & +1 & -1 & -1 & +1 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\frac{1}{x}+\frac{4}{y} = \frac{1}{12}$, where $y$ is and odd integer less than $61$. Find the positive integer solutions (x,y). $\frac{1}{x}+\frac{4}{y} = \frac{1}{12}$, where $y$ is and odd integer less than $61$.
Find the positive integer solutions (x,y).
| Let, $y=2n-1$
given, $2n-1\lt61$, $2n \lt 62$
$\frac{1}{x}+\frac{4}{y}=\frac{1}{12}$
$\frac{1}{x}+\frac{4}{2n-1}=\frac{1}{12}$
$\frac{1}{x}=\frac{1}{12}-\frac{4}{2n-1}$
$\frac{1}{x}=\frac{2n-1-48}{(12)(2n-1)}$
$\frac{1}{x}=\frac{2n-49}{(12)(2n-1)}$
$x=\frac{(12)(2n-1)}{2n-49}$
$x=\frac{(1)(3)(4)(2n-1)}{2n-49}$
By caref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/182882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Finding a basis for the solution space of a system of Diophantine equations Let $m$, $n$, and $q$ be positive integers, with $m \ge n$.
Let $\mathbf{A} \in \mathbb{Z}^{n \times m}_q$ be a matrix.
Consider the following set:
$S = \big\{ \mathbf{y} \in \mathbb{Z}^m \mid \mathbf{Ay} \equiv \mathbf{0} \pmod q \big\}$.
It ... | Well, here is how it works over a field. We take $q=5.$ We will start with $A$ as a 2 by 4,
$$ A \; = \; \left( \begin{array}{cccc}
2 & 3 & 4 & 1 \\
3 & 4 & 0 & 1
\end{array} \right) $$
We begin a sequence of elementary row operations, first multiply the first row times 3 and the second by 2,
$$ \left( \begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Solving a literal equation containing fractions. I know this might seem very simple, but I can't seem to isolate x.
$$\frac{1}{x} = \frac{1}{a} + \frac{1}{b} $$
Please show me the steps to solving it.
| $\frac{1}{x} = \frac{b}{ab} + \frac{a}{ab}$
$\frac{1}{x} = \frac{a + b}{ab}$
$x = \frac{ab}{a + b}$
note that $\frac{1}{x} = \frac{1}{a} + \frac{1}{b}$ is possible if and only if $\frac{1}{a} + \frac{1}{b} \neq 0$. This implies that $a \neq -b$; and, hence $a + b \neq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/184709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Simplify $\tan^{-1}[(\cos x - \sin x)/(\cos x + \sin x)]$
Write the following functions in simplest form:
$$\tan^{-1}\left(\frac{\cos(x)-\sin(x)}{\cos(x)+\sin(x)}\right), \quad 0<x<\pi$$
Please help me to solve this problem. I have been trying to solve this from last 3 hours. I can solve simple inverse trigonomet... | It is $\tan ^{-1}\dfrac{\cos \left(x\right)-\sin \left(x\right)}{\cos \left(x\right)+\sin \left(x\right)}$
We will divide in bracket with cosx to get it in tan form which will be easy for us to simplify
$\implies \tan ^{-1}\dfrac{\dfrac{\cos \left(x\right)-\sin \left(x\right)}{\cos \left(x\right)}}{\dfrac{\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Finding the sum of this alternating series with factorial denominator. What is the sum of this series?
$$ 1 - \frac{2}{1!} + \frac{3}{2!} - \frac{4}{3!} + \frac{5}{4!} - \frac{6}{5!} + \dots $$
| Maybe one can do it without using power series:
$$
\begin{align}
\sum_{n=0}^{\infty}(-1)^n\frac{n+1}{n!}
&=\sum_{n=0}^{\infty}(-1)^n\frac{n}{n!}+\sum_{n=0}^{\infty}(-1)^n\frac{1}{n!}\\
&=\sum_{n=1}^{\infty}(-1)^n\frac{1}{(n-1)!}+\sum_{n=0}^{\infty}(-1)^n\frac{1}{n!}\\
&=\sum_{k=0}^{\infty}(-1)^{k+1}\frac{1}{k!}+\sum_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to prove $ \int_0^\infty e^{-x^2} \; dx = \frac{\sqrt{\pi}}2$ without changing into polar coordinates? How to prove $ \int_0^\infty e^{-x^2} \; dx = \frac{\sqrt{\pi}}2$ other than changing into polar coordinates? It is possible to prove it using infinite series?
| Because $x \mapsto e^x$ is convex, $\displaystyle \left( 1- \frac{u}{n} \right)^n \leq e^{-u} \leq \left( 1+ \frac{u}{n} \right)^{-n}$, so $\displaystyle \int_0^{\sqrt{n}} \left( 1- \frac{x^2}{n} \right)^n dx \leq \int_0^{\sqrt{n}} e^{-x^2} dx \leq \int_0^{\sqrt{n}} \left( 1+ \frac{x^2}{n} \right)^{-n} dx$.
We use the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 7,
"answer_id": 0
} |
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.