Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$ How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$
I tried to factor and I got $2\cos^4(x)+(-2\sin^2(x)-3)(\cos^4(x)+\sin^4(x))$ but that doesn't lead me to my goal.
I also tried to write all the cosines i... | Apply $a^2+b^2=(a+b)^2-2ab$ in $$\sin^4x+\cos^4x=(\sin^2x)^2+(\cos^2x)^2$$
and $a^3+b^3=(a+b)^3-3ab(a+b)$ in $$\sin^6x+\cos^6x=(\sin^2x)^3+(\cos^2x)^3$$
Finally the re-arrange the terms
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/742358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Visually deceptive "proofs" which are mathematically wrong Related: Visually stunning math concepts which are easy to explain
Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually wrong. (e.g. missing square puzzle)
Do you know the other examples... | This is my favorite.
\begin{align}-20 &= -20\\
16 - 16 - 20 &= 25 - 25 - 20\\
16 - 36 &= 25 - 45\\
16 - 36 + \frac{81}{4} &= 25 - 45 + \frac{81}{4}\\
\left(4 - \frac{9}{2}\right)^2 &= \left(5 - \frac{9}{2}\right)^2\\
4 - \frac{9}{2} &= 5 - \frac{9}{2}\\
4 &= 5
\end{align}
You can generalize it to get any $a=b$ that you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/743067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "137",
"answer_count": 25,
"answer_id": 7
} |
Find the length of... Find the length of $\overline{AD}$ knowing that it is divided into three equal parts by to tangent circles with radius respectively $3\sqrt{3}$ and $\sqrt{3}$ . Here's the graph: so the segments $\overline{AB}$, $\overline{BC}$ and $\overline{BC}$ are equal.
Thank you very much!
|
Let $AB=BC=CD=x$.
$O1M$, $O2E$ $⊥AD$. (Perpendicular bisectors)
$O1MEO2$ - rectangular trapezoid.
$AM=MB$ and $CE=ED$. Hence $ME = 2x$.
From right triangles $O1AM$ and $O2ED$ we got $O1M = \sqrt{R^2 - \frac{x^2}{4}}$ and $O2E = \sqrt{r^2 - \frac{x^2}{4}}$
Use Pythagorean theorem for triangle $O1FO2$ ($O2F ⊥ O1M$):
$(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/745123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $Y_1, Y_2,\ldots,Y_n$ denote a random sample from the uniform distrib... Help find finding $ \text{Var}\left[\hat{\theta}_{2}\right]$ Let $Y_1, Y_2,\ldots,Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$
Find the efficiency of... | Variance can be evaluated as follows
$$
\text{Var}\left[\hat{\theta}_{2}\right]=\text{Var}\left[Y_{(n)}-\frac{n}{n+1}\right]=\text{Var}\left[Y_{(n)}\right]=\text{E}\left[Y_{(n)}^2\right]-\left(\text{E}\left[Y_{(n)}\right]\right)^2.
$$
First, we calculate $\text{E}\left[Y_{(n)}^2\right]$. Using the result from here, we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/750703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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The number of primes in the factorization of $N!$ Is there an approximation to the number of primes in the factorization of $N!$?
For example:
*
*For $N=10$, this number is $15$.
*For $N=100$, this number is $239$.
*For $N=1000$, this number is $2877$.
*For $N=10000$, this number is $31985$.
*For $N=100000$, thi... | This does not completely answer your question but is helpful to know and the argument can be made completely rigorous.
The highest power of a prime $p$ dividing $N!$ is
$$\left\lfloor \dfrac{N}p \right\rfloor + \left\lfloor \dfrac{N}{p^2} \right\rfloor + \cdots \sim \dfrac{N}{p-1}$$ and the number of primes less than $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/751752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Prove the inequality.Let a, b and c be nonnegative real numbers. Let $a$, $b$ and $c$ be nonnegative real numbers. Prove that $a^4+b^4+c^2\ge 8^{½}abc$
| Another way:
$$
a^4+b^4+\frac{c^2}{2}+\frac{c^2}{2}\geq 4\left(a^4b^4\frac{c^2}{2}\frac{c^2}{2}\right)^\frac 14=2^\frac 32 abc.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/753719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How many 6 letter words can be made with these conditions? The letters that can be used are A, I, L, S, T.
The word must start and end with a consonant. Exactly two vowels must be used. The vowels can't be adjacent.
| There are $3$ ways to choose the first consonant, and $3$ ways to choose the last one.
Of the four remaining letters, exactly two must be vowels. There are $$\binom{4}{2}-3=6-3=3$$ ways to choose which spots the vowels can be in, since they cannot be adjacent. Then, there are $2$ choices for the first vowel, and $2$ ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/754888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Lagrange multipliers from hell I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way:
"Find the closest and furthest points on the circle made from the intersection of the ball $(x-1)^2+(y-2)^2+(z-3)^2=9$ and the plane $x-2z=0$ from the point $(0,0)$".
What I di... | $\newcommand{\e}{\mathbf{e}}$Here's an algebraic approach:
Let $P$ denote the plane with equation $x - 2z = 0$ and $S$ the sphere
$$
(x - 1)^{2} + (y - 2)^{2} + (z - 3)^{3} = 9.
$$
The vectors $\e_{1} = (2, 0, 1)/\sqrt{5}$ and $\e_{2} = (0, 1, 0)$ are an orthonormal basis of $P$. The orthogonal projection to $P$ of $c ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/755510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Calculate $\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$ I am trying to calculate:
$$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$
I am not looking for an answer but simply a nudge in the right direction. A strategy, just something that would get me started.
So, after doing the Taylor Expansion on the $\ln(1-x+x^2)$ ig to the follo... | Differentiation under the integral sign can be applied to this.
Consider
\begin{align}
I(a)&=\int_0^1 \, \frac{\ln{(1-a\,(x-x^2))}}{x-x^2}\, dx \tag 1\\
\frac{\partial}{\partial a} I(a)&=\int_0^1\, -\frac{1}{1-a\,(x-x^2)}\, dx \\
&= -\frac{4 \, \sqrt{-y^{2} + 4 \, y} \arctan\left(\frac{\sqrt{-y^{2} + 4 \, y}}{y - 4}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/756598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
} |
Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
| Let
$A=\begin{bmatrix}
a & b & b & b & \ldots & b \\
b & a & b & b & \ldots & b \\
b & b & a & b & \ldots & b \\
b & b & b & a & \ldots & b \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
b & b & b & b & \ldots & a
\end{bmatrix}$
a $n\times n$ matrix, cause adding/substracting a scalar multiple of a row to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/757320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How prove this $a^n-b^n$ always have prime factor $P$ and $P>n$
Let $p_{1},p_{2},p_{3}$ be different prime numbers, and let the positive integer $n$, be defined by
$$n=p_{1}p_{2}p_{3}.$$
Show that:
For any two positive integer $a,b$ ,then $a^n-b^n$ always has a prime factor $P$ satisfying $P>n$
This problem is f... | If $a=b$ then $a^n-b^n=0$ is divisible by any prime, so we are done since there are infinitely many primes.
Otherwise we may WLOG assume $a>b$. Let $d=\gcd(a, b)$. By Zsigmondy's theorem, $\left(\frac{a}{d}\right)^n-\left(\frac{b}{d}\right)^n$ has a prime factor $p$ that does not divide $\left(\frac{a}{d}\right)^k-\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/757819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Range of f(x) = $\frac{\sqrt3\,\sin x}{2 + \cos x}$ Can you give any idea about the range of the following function?
$$f(x) = \frac{\sqrt{3}\,\sin x}{2 + \cos x}$$
| Consider $f^2(x) = \dfrac{3 - 3cos^2x}{cos^2x + 4cosx + 4}$. Now let $t = cosx$, then look at $f(t) = \dfrac{3 - 3t^2}{t^2 + 4t + 4}$. We find $f_{max}$. $f'(t) = \dfrac{-6(2t + 1)}{t + 2} = 0$ when $t = \dfrac{-1}{2}$. So $f(-1) = 0$, $f(\frac{-1}{2}) = 1$, and $f(1) = 0$. So $f_{max}^2 = 1$. This means that: $f^2(x) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/762243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Successively longer sums of consecutive Fibonacci numbers: pattern? Consider the following:
$$\begin{align}
F_{n-1}+F_{n-2}&=F_n\\
F_{n-1}+F_{n-2}+F_{n-3}&=F_{n-1}+F_{n-1}\\
&=2F_{n-1}\\
F_{n-1}+F_{n-2}+F_{n-3}+F_{n-4}&=F_n+F_{n-2}\\
&=L_{n-1}\\
F_{n-1}+F_{n-2}+F_{n-3}+F_{n-4}+F_{n-5}&=F_{n-1}+L_{n-2}\\
&=F_{n-1}+F_{n-... | $$F_n = \left(\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n\right)_{2,1}$$
which let's us treat it like any other summation except using matrices instead of scalar numbers, just remember that matrix multiplication doesn't commute and everything else is the same.
Using $X = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/765104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Find the monic generator of and ideal. Let $\mathbb{F}$ be a subfield of complex numbers, and let
$$
A = \begin{bmatrix}
1 & -2 \\[0.1em]
0 & 3 \\[0.1em]
\end{bmatrix}
$$
Find the monic generator of the ideal of all polynomials $f$ in $F[x]$ such that $f(A)=0.$
| We can solve this in a reasonably simple manner.
Consider some $f \in F[x]$ such that $f(A) = 0$. Suppose that $deg(f) = 1$. Then, $f$ is of the form $$ f(x) = cx + d. $$ However, we need $f(A)$ to be zero and all values are in $\mathbb R^{2x2}$, so
\begin{align*}
f(A) &= cA + d \\
&= c\begin{bmatrix}1&-2\\0&3\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/766145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Formula for $\sum_{k=0}^n k^d {n \choose 2k}$ If $d \geq 1$ is an integer, is there a general formula for $$\sum_{k=0}^n k^d {n \choose 2k}\,?$$
We know that $\sum_{k=0}^n k {n \choose 2k} = \frac{n2^n}{8}$ and $\sum_{k=0}^n k^2 {n \choose 2k} = \frac{n(n+1)2^n}{32}$.
Note that ${n \choose 2k} = 0$ when $2k > n$.
| The answer that I presented in my other post is more complicated than
it needs to be.
Suppose we seek to evaluate
$$\sum_{k=0}^n k^d {n\choose 2k}.$$
We observe that
$$k^d = \frac{d!}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{d+1}} \exp(kz) \; dz.$$
This yields for the sum
$$\frac{d!}{2\pi i}
\int_{|z|=\epsilon} \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/767146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Find the limit $\lim_{x \rightarrow 0} \frac{(1+x)^\frac{1}{8}-(1-x)^\frac{1}{8} }{x}$ I am trying to evaluate the following limit
$$\lim_{x \rightarrow 0} \frac{(1+x)^\frac{1}{8}-(1-x)^\frac{1}{8} }{x}$$
If we use the binomial expansion of the numerator term, the answer is $\frac{1}{4}$. The same answer is obtained i... | $$\lim_{x \rightarrow 0} \frac{(1+x)^\frac{1}{8}-(1-x)^\frac{1}{8} }{x} =\lim_{x \rightarrow 0} \frac{(1+x)^\frac{1}{8}-1+1-(1-x)^\frac{1}{8} }{x}=\lim_{x \rightarrow 0} \frac{(1+x)^\frac{1}{8}-1 }{x}+\lim_{x \rightarrow 0} \frac{(1-x)^\frac{1}{8}-1 }{-x}=\frac{1}{8}+\frac{1}{8}=\frac{1}{4} $$
We applied the known limi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/768531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Show the limit of the following is $\dfrac{1}{12}$
Show that $$\lim_{n\to \infty}n^2 \log \left(\dfrac{r_{n+1}}{r_{n}} \right)=\dfrac{1}{12}$$ where $r_{n}$ is defined as;
$$r_{n}=\dfrac{\sqrt{n}}{n!} \left(\dfrac{n}{e} \right)^n$$.
Now I simplified $\dfrac{r_{n+1}}{r_{n}}$ to $$\dfrac{r_{n+1}}{r_{n}}= \left(\dfr... | Application of the rule of L'Hospital is a good idea:
$$\lim_{n\to \infty} \dfrac{n \log \left(\dfrac{n+1}{n} \right) +0.5 \log \left(\dfrac{n+1}{n} \right)-1}{\dfrac{1}{n^2}} =\lim_{x\to 0} \dfrac{\dfrac{1}{x} \log(1+x) +0.5 \log (1+x)-1}{x^2} =(L'H) \lim_{x\to 0} \dfrac{\dfrac{-1}{x^2} \log(1+x)+\dfrac{1}{x(1+x)}+\df... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/769805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to integrate $\int\frac1{1+2x}dx$? What approach would be ideal in finding the solution to the integration problem $\int\frac1{1+2x}dx$?
| In general
\begin{align}
\int\frac{a}{bx+c}dx=\frac{a}{b}\ln (bx+c)+C
\end{align}
Proof:
Let $u=bx+c$ then $x=\cfrac{u-c}{b}$ and $dx=\cfrac{du}{b}$. Hence
\begin{align}
\int\frac{a}{bx+c}dx=\int\frac{a}{u}\cdot\frac{du}{b}=\frac{a}{b}\int\frac{du}{u}=\frac{a}{b}\ln u+C=\frac{a}{b}\ln (bx+c)+C
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/770792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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series convergence proof. Use Theorem 8.8 to show that if $0<x\leq 1$ then ln$(1+x)=\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}x^k$.
Theorem 8.8 states that for each natural number $n$ and each number $x>-1$ there is a number $c$ strictly between 0 and $x$ such that ln$(x+1)=x-\frac{x^2}{2}+..+\frac{(-1)^{n+1}x^n}{n}+\f... | I think you are on the right track. I reorganize your proof here, and add one or two details. For $x \in (0, 1]$, let $S_n(x) = \displaystyle \sum_{k=1}^n \dfrac{(-1)^{k+1}x^k}{k}$, then you need to prove: $\displaystyle \lim_{n \to \infty} \left(S_n(x) - ln(1+x)\right) = 0$, and using theorem 8.8, we have: $S_n(x) - l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/770977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
using substitution wrongly Solving integral, first way:
$$\int \frac{du}{u^2-9}=-\int \frac{du}{9-u^2}$$
$$u={3\sin v}$$
$$du=3\cos vdv$$
$$-\int \frac{3\cos vdv }{9-9\sin^{2}v}=-\frac 13\int\frac{dv}{\cos v}=-\frac 13\ln\left(\sec v+\tan v\right)=-\frac13 \ln \frac {\frac u3}{\sqrt{1-\frac{u^2}{9}}}$$
$$-\frac13 \ln \... | As Grid commented, before any simplification
$$\ln\left(\sec v+\tan v\right)=\log \left(\frac{u}{3 \sqrt{1-\frac{u^2}{9}}}+\frac{1}{\sqrt{1-\frac{u^2}{9}}}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/771413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Max. value $M$ of $c^2$ for which points lie opposite of $x(a+b)^2-(ab+bc+ca+1)y+2 = 0$ Let $M$ be the the maximum value of $c^2$ for which $O(0,0)$ and $A(1,1)$ does not lie on
opposite side of straight line $x(a+b)^2 -(ab+bc+ca+1)y +2=0.$Then value of $M$ is
$\bf{My\; Solution::}$ Given $O(0,0)$ and $A(1,1)$ does no... | Hint:
\begin{align}
(a+b)^2-ab -bc - ca +1 &\ge (a+b)^2 - \tfrac12(a^2+b^2) - \tfrac16(9b^2+c^2)-\tfrac16 (c^2+9a^2)+1 \\
&= -(a-b)^2+1 - \tfrac13 c^2 \\
&\ge 1- \tfrac13 c^2
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/772382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Squares constructed externally on the sides of a triangle and concurrent lines On the sides $BC, CA$ and $AB$ of the triangle $ABC$ we construct externally the squares $BCDE, ACFG $ and $ABHI$. Denote $A', B'$ and $C'$ the intersectiond points of the lines $BF$ and $CH$, $AD$ and $CI$, respectively $AE$ and $BG$. Prove... | The lines $AA',BB',CC'$ are the altitudes of the triangle, so they intersect in the orthocenter.
To show that these are in fact altitudes, you can use a computation on coordinates. W.l.o.g. assume the following coordinates:
$$A=\begin{pmatrix}-1\\0\end{pmatrix}\qquad
B=\begin{pmatrix}1\\0\end{pmatrix}\qquad
C=\begin{pm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/774381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate the inequality $a,b,c$ are the sides of a triangle.Then show that
$$(a+b+c)^3 > 27(a+b-c)(b+c-a)(c+a-b)$$ Also give the case where equality holds i.e. $$(a+b+c)^3=27(a+b-c)(b+c-a)(c+a-b)$$
I tried triangle inequality, AM-GM inequality in many way but cannot help my cause.
| First apply AM-GM inequality: $(a+b+c)^3 \geq 27abc$, then you need to prove: $abc \geq (a+b-c)(b+c-a)(c+a-b)$, and this follows from:
$a^2 \geq a^2 - (b-c)^2$
$b^2 \geq b^2 - (c-a)^2$
$c^2 \geq c^2 - (a-b)^2$.
by multiply all three inequalities and taking square root.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/775288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Binomial coefficients (Concrete mathematics 5.39)
Show that if $xy = ax+by$ then $$x^ny^n = \sum_{k=1}^n \binom{2n-1-k}{n-1} (a^nb^{n-k}x^k + a^{n-k}b^ny^k)$$ for all $n>0$. Find a similar formula for the more general product $x^my^n$. (There formulas give useful partial fraction expansions, for example when $x=1/(z-c... | (a) We prove it by induction. Initial case, $n=1$:
\begin{align}
\sum_{k=1}^n{\binom{2n-1-k}{n-1} (a^nb^{n-k}x^k + a^{n-k}b^ny^k)} & = \binom{0}{0}(a^1b^0x^1 + a^ob^1y^1) \\
& = ax+by \\
& = x^ny^n.\qquad\checkmark
\end{align}
Now assume the statement is true for some $n\gt 0$ and consider the case for $n+1$.
\begin{al... | {
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"url": "https://math.stackexchange.com/questions/777931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Group theory, conjugation of permutations I have a past exam question that says...
Decompose the following permutations into a product of disjoint cycles. Are the two permutations conjugate?
$$\alpha= \begin{bmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 &9\\
7 & 4 & 5 & 3 & 8 & 6 & 9 & 1 & 2\\
\end{bmatrix}$$
and
... | Decomposition in cycle is fairly easy, and it yields
$$\alpha = (1\ 7\ 9\ 2\ 4\ 3\ 5\ 8)$$
$$\beta = (1\ 6\ 2\ 7\ 3\ 8\ 4\ 9\ 5)$$
Notice $6$ does not appear in the cycle decomposition of $\alpha$ (it would be a cycle of length 1, usually they are not written).
Two permutations of $S_n$ are conjugate iff they have the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/778350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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How prove this $ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. $ infinitely many special numbers Qustion
A special number is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with $$ n = \dfrac {a^3 + 2b^3} {c^3 + 2d^3}. $$ Prove that
i) there are infinitely many special numbers;
ii) $20... | First note that $-2$ is not a cubic residue modulo $19$. To see this, suppose
$x^3\equiv -2\pmod{19}$. Then $x^{18}\equiv (-2)^6 \equiv 64 \equiv 7\pmod{19}$
Clearly $19$ does not divide $x$, or we would have $x^3\equiv 0\pmod{19}$, and so by Fermat's Little Theorem we obtain $x^{18}\equiv 1\pmod{19}$, which is a contr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/782297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove that $\lim_{n\to\infty} (\sqrt{n^2+n}-n) = \frac{1}{2}$ Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$
Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it sp... | From $a^2-b^2=(a+b)(a-b)$ we have
$$\sqrt{n^2+n}-n=\frac{n^2+n-n^2}{\sqrt{n^2+n}+n}=\frac{n}{\sqrt{n^2+n}+n}=
\frac{1}{\sqrt{1+\frac{1}{n}}+1}$$
from which the result follows immediately.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/783536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 10,
"answer_id": 3
} |
Halving a tiny angle by doubling a side of a triangle. You have a triangle $ABC$ with a right angle $\angle CAB$. The line stretch between $A$ and $B$ shall be very long in comparison to the other sides. Now you are going to measure the angle $\angle ABC$ which is very tiny, e.g. $1'$.
Now you keep the line stretch be... | The rough idea is as Vizzy says: for small angles $\theta$, $\tan \theta \approx \theta$.
We can make this rigorous.
Suppose the original angle is $\theta$, and that originally $AC = l$ and $AB = L$
($L \gg l$).
Then
$$
\tan \theta = \frac{l}{L}
$$
so after doubling the length of $AB$, the new angle $\theta_1$ will sat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/783859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Find $[E(Y^2)]$. When $Y = 3 * X - 5$ and $X$ is distributed in range $[0, 5]$ Need help with exercise.
Random variable $X$ is evenly distributed in range $[0, 5]$. Need to find $E[Y^2]$ when $Y = 3X - 5$
Every hint/tip will be appreciated.
Thank you
Alternative solution
$$E[Y^2] = E[(3X-5)^2] = E[9X^2-30X+25] = 9E[X^2... | I assume that $X$ is uniformly distributed. The CDF of $X\sim\mathcal{U}(0,5)$ is
$$
F_X(x)=\Pr[X\le x]=\frac{x-0}{5-0}=\frac x5.
$$
Therefore
$$
\begin{align}
\Pr[Y\le y]&=\Pr[3X-5\le y]\\
F_Y(y)&=\Pr[3X\le y+5]\\
&=\Pr\left[X\le \frac{y+5}{3}\right]\\
&=F_X\left(\frac{y+5}{3}\right)\\
&=\frac{y+5}{15}.
\end{align}
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/783941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
value of $a+b$ of the following function $f(x)=x^3-3x^2+5x\;$ and
$\;f(a)=1,f(b)=5.\;$ Find $a+b$.
I know only one real root exist for each equation as derivative of the function is always positive .I do not intend to use the formula of roots of cubic equation. How should i go about this problem ????
| I assume that $a$ and $b$ are real numbers (this can be guaranteed since $f(a)=1$ and $f(b)=5$ have real solutions.
As you said that $f^\prime$ is always positive that means $f$ is a strictly increasing function. Since $f(0)=0$ then $0<a<b$.
Note that we have
$$f(\frac{a+b}2)-\frac{f(a)}2-\frac{f(b)}2=-\frac38(b-a)^2(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/785337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Parametric equations where sin(t) and cos(t) must be rational Suppose there are parametric equations
$$
x(t) = at - h\sin(t)
$$
$$
y(t) = a - h\cos(t)
$$
and it is required that both $\sin(t)$ and $\cos(t)$ should be rational.
What the values of $t$ should be in that case?
Thanks.
| Your question (at least how you wrote it) is really,
For what values of $t$ are both $\cos t$ and $\sin t$ rational?
Note that if $x$ and $y$ are real numbers with $x^2 + y^2 = 1$, then there exists a real number $t$ such that $x = \cos t$, $y = \sin t$.
Therefore, we want solutions to $x^2 + y^2 = 1$ for $x, y \in \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
how can we get from $x \equiv 4 \pmod 9$ to $x \equiv 1 \pmod 3$ I want to solve the system:
$$ x \equiv 1 \pmod 3 , x \equiv 2 \pmod 5, x \equiv 3 \pmod 7, x \equiv 4 \pmod 9, x \equiv 5 \pmod {11}$$
The numbers $3,5,7,9,11$ are not pairwise coprime,especially the numbers $3,9$ are not coprime.
According to my notes, ... | $ x \equiv 4 \mod 9 $ means $x = 9k + 4$ for some integer $k$. that means $x = 3(3k + 1) + 1$ and $3k+1$ is an integer which means $x \equiv 1 \mod 3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/788956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Calculate $\lim_{x \to \infty} x - \sqrt{x^2 + 2x}$ without derivations How can I calculate $\displaystyle \lim_{x \to \infty} x - \sqrt{x^2 + 2x}$?
Here is what I've done so far:
Multiplying by $\displaystyle \frac{x + \sqrt{x^2 + 2x}}{x + \sqrt{x^2 + 2x}}$
I got $\displaystyle \frac {-2x}{x+\sqrt{x^2 + 2x}}$
Multi... | Try:
$$
\frac{\sqrt{x^2 + 2x}}{x} = \sqrt{\frac{x^2 + 2x}{x^2}} = \sqrt{1 + \frac{2}{x}}
$$
By continuity of the square root function this means
$$
\lim_{x \to \infty} \frac{\sqrt{x^2 + 2x}}{x} = 1.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/789772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Understanding substitution to compute primitive To compute the primitive
$$\int x^2 \sqrt[3]{x^3+3}\ dx$$
I am trying this:
$t = x^3+3$
$dt = 3x^2dx$
but
$\int x^2 \sqrt[3]{x^3+3}\ dx=\ ?$
How to continue from here?
| Notice that $3x^2$ is the derivative of $x^3+2$.
So if you set $t=x^3+2$, you get $\dfrac{dx}{dt}=3x^2$ which gives $dx = 3x^2 dt$.
Now the whole thing becomes:
$$\int x^2 \sqrt[3]{x^3+3}\ dx = \frac{1}{3}\int 3x^2\sqrt[3]{x^3+3} \ dx = \frac13 \int t^{1/3} dt = \frac{1}{3}\frac{1}{1+\frac{1}{3}} t^{1+\frac{1}{3}} + C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/790405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
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Find the area of the surface of revolution generated by revolving about the $x$-axis the hypocycloid $x=a\cos^3\theta$, $y=a\sin^3\theta$ Find the area of the surface of revolution generated by revolving about the $x$-axis the hypocycloid $x=a\cos^3\theta$, $y=a\sin^3\theta$ ($0 \leq \theta \leq \pi$)
I know you have t... | The standard formula for rotation about the $x$-axis, through an angle of $2\pi$-radians, when $x$ and $y$ are given as functions of $\theta$ is
$$A = 2\pi \int_{\theta_1}^{\theta_2} y\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}\theta}\right)^{\!\!2} + \left(\frac{\mathrm{d}y}{\mathrm{d}\theta}\right)^{\!\!2}}~\mathrm{d}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/792101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find the area between the functions I have a homework where I have to find the area between the functions $y=x-2$ and $y=\sqrt{7-2x}$.
Computing the integral I have found $A=\frac {x^2}2-2x-\frac13 (7-2x)^{3/2}$; The upper limit is $\frac72$ and the downlimit is $2$.
After make the computing I have found $A= -0.60705... | First, you need to draw a picture:
The two curves cross, so you will need to divide this into two separate integrals. They cross when $x-2=\sqrt{7-2x}$, so that $(x-2)^2 = x^2-4x+4 = 7-2x$. Solving gives $x=3$ (and $x=-1$, which you don't care about here). The integrals of the two functions are
$$\int (x-2)\,dx = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/792175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $133\mid \left(11^{n+2}+12^{2n+1}\right)$ Prove $133\mid \left(11^{n+2}+12^{2n+1}\right)$, where $n$ is a non-negative integer.
So, I went about proving this using Fermat's theorem.
So I had $11^{n}\cdot 11^2+(12^2)^n= 0\ (\mod 133)$
then $11^n\cdot 11^2+1728^n=0\ (\mod133)$
and finally $1^n+132^n=0\ (\mod 133)... | since $133=7*19$, first show that $$7\, |\, \left(11^{n+2} + 12^{2n+1}\right)$$
then $$19\, |\, \left(11^{n+2} + 12^{2n+1}\right)$$
Since $gcd(7,19)=1$, the result follows.
$$ 11^{n+2}\equiv 11^{n}\cdot 11^2\equiv 2\cdot11^{n}\bmod 7$$
$$12^{2n+1}\equiv 5\cdot 144^{n}\equiv 5\cdot 11^{n}\bmod 7$$
So $$11^{n+1} + 12^{2n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/792264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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if $\frac{1}{1+x+f(y)}+\frac{1}{1+y+f(z)}+\frac{1}{1+z+f(x)}=1$ find the function $f(x)$ Find all functions $f(x):(0,\infty)\to(0,\infty) $satisfying
$$\dfrac{1}{1+x+f(y)}+\dfrac{1}{1+y+f(z)}+\dfrac{1}{1+z+f(x)}=1$$
whenever $x,y,z$ are positive numbers and $xyz=1$
I know this if
$$xyz=1\Longrightarrow \dfrac{1}{1+x+xy... | Consider a function $f$ that satisfies the proposed condition.
*
*Setting $(x,y,z)=(1,1,1)$ we see that $f(1)=1$.
*Now, consider $t>0$ and let $a=f(t)$, $b=f(1/t)$. Setting $(x,y,z)=(t,1/t,1)$ we get
$$
\frac{1}{1+t+b}+\frac{1}{2+1/t}+\frac{1}{2+a}=1\tag{1}
$$
and setting $(x,y,z)=(1/t,t,1)$ we get
$$
\frac{1}{1+1/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/795702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Constructing a polynomial with certain zeroes. I want to construct a polynomial $f(x)$ that has zeroes at $-9,\,-5,\,0,\, 5,\, 9$.
Can somebody provide a method (or perhaps some hints) for solving this?
| Zeroes at $-9$, $-5$, $0$, $5$, $9$... This means that the polynomial will be of the form:
$$f(x)=a(x)(x+9)(x+5)(x-5)(x-9)$$
Where $a$ is a constant. Any value of $a$ will work here. These are all correct answers:
$$f(x)=x(x+9)(x+5)(x-5)(x-9)$$
$$f(x)=\sqrt 2(x)(x+9)(x+5)(x-5)(x-9)$$
$$f(x)=\pi x(x+9)(x+5)(x-5)(x-9)$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/796072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Solving implicit equation for x or y I want to solve the following equation for $x$ or $y$ (does not matter wich one) analytically.
$$\sqrt[3]{x+y} + \sqrt[3]{x-y} = 1$$
Wolframalpha returns following solution, but I could not think of a way how to get there:
$$ x+1 \neq 0 \qquad y = \frac{(x+1) \sqrt{8 x-1}}{3 \sqr... | $$(\sqrt[3]{x+y} + \sqrt[3]{x-y})^3 = 1$$
$$x+y+3(\sqrt[3]{x+y}\cdot\sqrt[3]{x-y})\underbrace{(\sqrt[3]{x+y} + \sqrt[3]{x-y})}_1+x-y=1$$
$$2x+3(\sqrt[3]{x+y}\cdot\sqrt[3]{x-y})=1$$
$$3(\sqrt[3]{x+y}\cdot\sqrt[3]{x-y})=1-2x$$
$$3(\sqrt[3]{x^2-y^2})=1-2x$$
$$27(x^2-y^2)=(1-2x)^3$$
$$27y^2 = 8 x^3+15 x^2+6 x-1$$
$$27y^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/799562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the average value of a function! over a region! Completely related to this question:
Finding surface area of part of a plane that lies inside a cylinder???
Find the average value of the function $f(x, y, z) = x^2yz$ over $S$
How does one do that?? I would think it would be just taking an average over the area... | For the two linked problems, here are two views of a graph of the cylinder $ \ x^2 \ + \ y^2 \ = \ 3 \ $ and the oblique plane $ x \ + \ 2y \ + \ 3z \ = \ 1 \ $ :
In order to find the average value of $ \ f(x,y,z) \ = \ x^2 \ y \ z \ $ over the region $ \ S \ $ of the plane lying within the cylinder, it will help to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/799914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Evaluating the limit of $\lim_{x\to\infty}(\sqrt{\frac{x^3}{x+2}}-x)$. How do I evaluate this limit:
$$\lim_{x\to\infty}\left(\sqrt{\frac{x^3}{x+2}}-x\right)$$
I tried to evaluate this using rationalizing the denominator, numerator and L'Hospital rule for nearly an hour with no success.
| Rationalizing, observe that:
\begin{align*}
\lim_{x\to\infty}\left(\sqrt{\frac{x^3}{x+2}}-x\right)
&= \lim_{x\to\infty}\left(\sqrt{\frac{x^3}{x+2}}-x\right)\left(\dfrac{\sqrt{\dfrac{x^3}{x+2}} + x}{\sqrt{\dfrac{x^3}{x+2}} + x}\right) \\
&= \lim_{x\to\infty} \dfrac{\dfrac{x^3}{x+2} - x^2}{\sqrt{\dfrac{x^3}{x+2}} + x} \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/800009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 0
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Find the value of the integral $\int_0^{2\pi}\ln|a+b\sin x|dx$ where $0\lt a\lt b$ Find the value of the integral $$\int_0^{2\pi}\ln|a+b\sin x|dx$$ where $0\lt a\lt b$. What is the use of this inequality. I tried to integrate the integral by parts, but the integral of the 2nd term was quite messy.Please help.
| This is just for $0<b < a $
\begin{align*}
I &= \int_0^{2\pi}\log(a+\sqrt{a^2 - b^2}+be^{i\theta})d\theta + \int_0^{2\pi}\log(a+\sqrt{a^2 - b^2}+be^{-i\theta})d\theta\\
&= \int_0^{2\pi} \log \left(2a^2 +2a\sqrt{a^2 - b^2} + (a + \sqrt {a^2 - b^2}) b (e^{i\theta} + e^{-i\theta}) \right )d\theta\\
&= \int_0^{2\pi} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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How to integrate $\int_0^{\pi/2} \ \frac{\cos{x}}{\sqrt{1+\cos{x}}} \, \mathrm{d}x.$ I need to somehow evaluate the following:
$$ \int_0^{\pi/2} \ \dfrac{\cos{x}}{\sqrt{1+\cos{x}}} \, \mathrm{d}x. $$
Can anyone give me any hints/pointers? I've tried to use parts, and some feeble substitutions, but to no avail :(
Thank... | Hint: $$\frac{1+\cos x}{2} = \cos^2\left(\frac x2\right)\Rightarrow\\\int\frac{\cos x}{\sqrt{1+\cos x}} \, \mathrm{d}x=\\\int\frac{\cos x}{\sqrt{2}\cos\left(\frac x2\right)} \, \mathrm{d}x=\\\int \frac{2\cos^2\left(\frac x2\right)-1}{\sqrt{2}\cos\left(\frac x2\right)} \, \mathrm{d}x=\\\int \sqrt{2}\cos\left(\frac x2\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Prove homogeneous inequality Prove that for all $a,b,c>0$ we have $$\frac{ab}{(a+b)^2}+\frac{bc}{(b+c)^2}+\frac{ac}{(a+c)^2}\leq\frac{1}{4}+\frac{4abc}{(a+b)(b+c)(c+a)}.$$
Please help me prove this homogeneous inequality.
| Let
$$
D=\frac{1}{4}+\frac{4abc}{(a+b)(b+c)(c+a)}-
\Bigg(\frac{ab}{(a+b)^2}+\frac{bc}{(b+c)^2}+\frac{ac}{(a+c)^2}\Bigg) \tag{1}
$$
We must show that $D$ is nonnegative. It will suffice to show
that
$$
D=\frac{1}{4}.\frac{(b-a)^2(c-a)^2(c-b)^2}{(a+b)^2(b+c)^2(c+a)^2} \tag{2}
$$
But this follows easily enough from the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/802849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Partial fraction integral Question:
$\int \dfrac{5 }{(x+1) (x^2 + 4) } dx $
Thought process:
I'm treating it as a partial fraction since it certainly looks like one.
I cannot seem to solve it besides looking at it in the "partial fraction" way.
My work:
1) Focus on the fraction part first ignoring the $\int $ and $d... | Note as in the comments you will have
$$
\dfrac{5 }{(x+1) (x^2 + 4) }
= \frac{A}{x+1} + \frac{Bx + C}{x^2+4}
$$
Cross multiplying and solving the equations gives
$$
\dfrac{5 }{(x+1) (x^2 + 4) }
= \frac{1}{x+1} - \frac{x - 1}{x^2+4}
$$
Which is easier to integrate. A sneakier way than solving the equations is the foll... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
log and poisson-like integral Here is a fun looking one some may enjoy.
Show that:
$$\int_{0}^{1}\log\left(\frac{x^{2}+2x\cos(a)+1}{x^{2}-2x\cos(a)+1}\right)\cdot \frac{1}{x}dx=\frac{\pi^{2}}{2}-\pi a$$
| Denote
$$
I(r)
=\int_{0}^{1}\log\left(\frac{x^{2}+2x r +1}{x^{2}-2x r+1}\right)\cdot \frac{1}{x}dx
$$
then
$$
\begin{align}
\frac{dI}{dr}
&=\int_0^1 \frac{4 \left(x^2+1\right)}{\left(2-4 r^2\right) x^2+x^4+1} dx\\
&=\int_0^1 \left(\frac{2}{x^2+2rx+1}+\frac{2}{x^2-2 r x+1} \right)dx\\
&=\frac{2 \tan ^{-1}\left(\frac{x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/808144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Evaluate $\int_0^\infty\frac{\ln x}{1+x^2}dx$ Evaluate $$\int_0^\infty\frac{\ln x}{1+x^2}\ dx$$
I don't know where to start with this so either the full evaluation or any hints or pushes in the right direction would be appreciated. Thanks.
| Another general approach :
Consider
$$
\int_0^\infty\frac{x^{b-1}}{a^2+x^2}\ dx.\tag1
$$
Rewrite $(1)$ as
\begin{align}
\int_0^\infty\frac{x^{b-1}}{a^2+x^2}\ dx&=\frac1{a^2}\int_0^\infty\frac{x^{b-1}}{1+\left(\frac{x}{a}\right)^2}\ dx.\tag2
\end{align}
Putting $x=ay\;\color{blue}{\Rightarrow}\;dx=a\ dy$ yields
\begin{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/808678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "50",
"answer_count": 7,
"answer_id": 0
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Does every integer $n > 2$ have an arithmetic expression involving at least two consecutive integers but excluding $n$ itself? For example:
$10 = 1 + 2 + 3 + 4$
$11 = 1 - 2 + 3 \times 4$
$12 = 3 \times 4$
$13 = -(1 - 2) + 3 \times 4$
$14 = 2 + 3 + 4 + 5$
$15 = 1 + 2 + 3 + 4 + 5$
$16 = (2/3)(4/5)(6 + 7 + 8 + 9)$
$17 = 1... | Every integer $n$ can be written as
$$n= (n+1)+(n+2)-(n+3)$$
where $(n+1)$ and $(n+2)$ are of course two consecutive integers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/810224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the value of $\sin {\frac{31 \pi}3}$ The task was to find out the value of $$\sin\frac{31\pi}3$$
This is a example in my book in which following steps are shown:
$$\begin{align}
\sin\frac{31\pi}3& =\sin\left(10\pi +\frac\pi3\right)\\
&=\sin\frac\pi3\\
&=\frac{\sqrt3}2
\end{align}$$
I cannot understand step 3, $... | You can confirm this by using the angle-sum formula:
$$\begin{align} \sin(10\pi + \pi/3) & = \sin(10\pi)\cos(\pi/3) + \cos(10 \pi)\sin(\pi/3) \\ \\ & = 0\cdot\frac 12 + 1 \cdot \sin (\pi/3) \\ \\ & = \sin(\pi/3)\\ \\ & = \sqrt 3/2\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/812611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
Sudoku puzzle with exactly 3 solutions While published sudoku puzzles typically have a unique solution, one can easily conceive of a sudoku puzzle with two solutions. However, is it possible to construct a sudoku puzzle with exactly 3 different solutions?
Inspired by https://puzzling.stackexchange.com/a/6.
| You could use this as a model (this works for $4 \times 4$ but needs to be adapted for $9\times 9$):
$\begin{array}{cccc}
1 & 2 & & \\
3 & 4 & 1 & 2 \\
& 1 & & \\
& 3 & & 1 \end{array} $
With solutions
$\begin{array}{cccc}
1 & 2 & 3 & 4 \\
3 & 4 & 1 & 2 \\
2/4 & 1 & 4/2 & 3 \\
4/2 & 3 & 2/4 & 1 \e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/813444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Multiplication of odds vs. multiplication of probabilities I always believed that probabilities could be multiplied, until I encountered a statement in Machine Learning by Peter Flach about odds:
"Bayes’ rule tells us that we should simply multiply them: 1:6 times 4:1 is 4:6, corresponding to a spam probability of $0.4... | Suppose that the prior probability of a message being spam is $\dfrac{1}{7}$, and the conditional probability of a message triggering the spam filter is $k$ if it is spam but $\dfrac{k}{4}$ if it is not spam for some $k$.
Then, given the spam filter is triggered, the posterior probability the message is spam is $\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/814482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
if A diagonalizable then show that $a=0$ Let $A= \begin{pmatrix}
2 & 0 & 0\\
a & 2 & 0 \\
b & c & -1\\
\end{pmatrix}$
if A diagonalizable then show $a=0$.
$P_A(x)=|xI-A|=(x-2)^2(x+1)=x^3-3x^2+4$
since A diagonalizable there is a P matrix such as $A=P^{-1}DP$
$D= \begin{pmatrix}
2 & 0 & 0\\
0... | Check whether there are 2 different eigenvectors to $2$. An $2$-eigenvector $v$ satisfies
$$
\begin{pmatrix}
0 & 0 & 0\\
a & 0 & 0 \\
b & c & -3
\end{pmatrix}
\begin{pmatrix}
v_1 \\ v_2 \\ v_3
\end{pmatrix}=0.
$$
This has two independent solutions, iff the above matrix has rank 1, which is iff $a=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/814792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is it possible to find the limit without l'Hopital's Rule or series Is it possible to find $$\lim_{x\to0}\frac{\sin(1-\cos(x))}{x^2e^x}$$
without using L'Hopital's Rule or Series or anything complex but just basic knowledge (definition of a limit, limit laws, and algebraic expansion / cancelling?)
| \begin{align}
\lim_{x \to 0} \frac{\sin(1 - \cos x)}{\mathrm{e}^x \cdot x^2}
&= \lim_{x \to 0} \frac{\sin(1 - \cos x)}{1 - \cos x}
\cdot \frac{1 - \cos x}{x^2} \cdot \mathrm{e}^{- x} \\
&= \lim_{u \to 0} \frac{\sin u}{u}
\cdot \lim_{x \to 0} \frac{1 - \cos x}{x^2}
\cdot \lim_{x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Use Lagrange Multipliers to determine max and min Using Lagrange Multipliers, determine the maximum and minimum of the function
$f(x,y,z) = x + 2y$
subject to the constraints
$x + y + z = 1$ and $y^2 + z^2 = 4$:
Justify that the points you have found give the maximum and minimum of $f$.
So,
$$
\nabla f = (λ_1)\nabla ... | We can find the extreme values using one constraint only: $z = 1 - x - y$, so $y^2 + (1-x-y)^2 = 4$.
Thus: $g(x,y) = y^2 + 1 + x^2 + y^2 - 2x - 2y + 2xy - 4 = 0$.
So: $\nabla f = \lambda\nabla g$ gives:
$(1,2) = \lambda(2x-2+2y,4y-2+2x)$. Thus:
$\dfrac{1}{2x-2+2y} = \dfrac{2}{4y-2+2x}$. Hence:
$4y-2+2x = 4x - 4 + 4y$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/827731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
formula for number triangles Hi, I have a triangle starting from $0$ and going up by one on the bottom row until there are $r$ items on the bottom row and there are $r$ rows a number is formed by adding the two numbers towards the fat end of the triangle together.
Example:
$$x \downarrow\ $$
$$
\begin{matrix}12\en... | A recursive solution: number the rows $0$ to $r-1$ with $0$ being the fattest, notice that the difference between consecutive numbers in row $k$ is $2^k$.
Denote by $f(n)$ the number in the last row of a triangle with $n$ rows. Then $f(n)=2f(n-1)+2^{n-2}$
Notice $f(1)=0,f(2)=1,f(3)=4$
From this recurrence we pass to $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to find the minimum/maximum distance of a point from elipse I have the point $(1,-1)$
and the ellipse $$x^2/9 + y^2/5 = 1 $$
How to find the minimum and maximum distance of the point from the ellipse ?
from exploring the ellipse I know that $$a = 3$$ , $$b =\sqrt{5}$$
$$ c = \sqrt{a^2-b^2} =\sqrt{9-5} = \sqrt{4}=2$... | The lack of symmetry in the geometric arrangement doesn't work in our favor, but fortunately the coefficients in the equations are only unpleasant to work with near the very end. As indicated by Padmanabha P Simha, we can use the Lagrange-multiplier method to aid in starting our work. We can extremize the "distance-s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Prove this inequality: If $|x+3|< 0.5$, show that $|4x+13| < 3$ If $|x + 3| < 0.5$, show that $|4x + 13| < 3$
This is what I've got so far:
$|4x + 13| = |(x + 3) + (3x + 10)|$
by the Triangle Inequality:
$|(x + 3) + (3x + 10)| \le |x + 3| + |3x + 10|$
Now I continue to apply the Triangle Inequality to reach:
$|(x + 3... | Yes, your solution is correct. A quicker approach could go like this: $|4x+13|=|4(x+3)+1|\leq 4|x+3|+1<4\cdot\frac12+1=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/832063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Mass of ellipsoid
We are given the elipsoid $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} \leq 1$$ with density function $$p(x,y,z)=x^2+y^2+z^2$$ Find the mass of the elipsoid.
I used the transformation $x=ar\sin\theta \cos\phi$, $y=br\sin \theta \sin \phi$ and $z=cr\cos\theta$. Spherical coordinates but I multi... | $$\int_{0}^{2\pi} \int_{0}^{\pi}\int_{0}^{1}(a^2r^2\sin^2\theta \cos^2\phi+b^2r^2\sin^2 \theta \sin ^2\phi+c^2r^2\cos^2 \theta)abcr^2\sin\theta drd\theta d\phi=$$
$$\int_{0}^{2\pi} \int_{0}^{\pi}\int_{0}^{1}a^3bcr^4\sin^3\theta \cos^2\phi drd\theta d\phi+\int_{0}^{2\pi} \int_{0}^{\pi}\int_{0}^{1}ab^3cr^4\sin^3 \theta \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/832985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
The Geodesics of a Sphere. I need to find the geodesics of a sphere. Then in polar coordinates
$$x=a \sin\theta \cos\phi \\y=a \sin\theta \sin\phi\\
z=a\cos\theta$$
Then $ds^2=dx^2+dy^2+dz^2$.
Can someone please tell me how is $ds^2=a^2(d\theta^2+\sin^2\theta \, d\phi ^2)$ obtained. I don't know much about spherica... | Rather than discuss via comments, here's an explicit list of steps/hints. Perhaps you can look at them only when stuck:
Step 1. Admittedly, the integral you mentioned is not easy. Here's one approach:
\begin{align*}
\phi &= \int \frac{k \, d\theta}{\sin \theta \sqrt{ \sin^2 \theta-k^2}}
= \int\frac{k \, d \theta}{\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/836090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
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How to prove inequality $\frac{a}{a+bc}+\frac{b}{b+cd}+\frac{c}{c+da}+\frac{d}{d+ab}\ge 2$ Question:
Let $$a,b,c,d>0,a+b+c+d=4$$
show that
$$\dfrac{a}{a+bc}+\dfrac{b}{b+cd}+\dfrac{c}{c+da}+\dfrac{d}{d+ab}\ge 2$$
when I solved this problem, I have see following three variables inequality:
Assumming that $a,b,c>0,a+b+c... | Take $1^{st}$ and $3^{rd}$ term apply the inequality, then take $2^{nd}$ and $4^{th}$ term and apply the inequality, you get
$$\ge 2\left(\frac1{1+\sqrt{bd}}+\frac1{1+\sqrt{ac}}\right)$$
again apply the inequality you get $\displaystyle \ge 2\left(\frac2{1+\sqrt{abcd}}\right)$
for $abcd =1$ you get your equality ie $a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/837565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
$\alpha$ and $\beta$ are solution of $a \cdot \tan\theta + b \cdot \sec\theta = c$ show $ \tan(\alpha + \beta) = \frac{2ac}{a^2 - c^2}$ If $\alpha$ and $\beta$ are the solution of $$a \cdot \tan\theta + b \cdot \sec\theta = c$$, then show that $$ \tan(\alpha + \beta) = \frac{2ac}{a^2 - c^2}$$
I did the following:
$$a ... | Consider the following:
\begin{align*}
c-a\tan \theta & = b \sec \theta\\
(c-a\tan \theta)^2 & = (b\sec \theta)^2\\
(a^2-b^2)\tan^2 \theta -2ac \tan \theta+(c^2-b^2) & = 0.
\end{align*}
Think of this as a quadratic in $\tan \theta$. It has two solutions $\tan \alpha$ and $\tan \beta$. So
\begin{align*}
\tan \alpha +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/838082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A Sine-Cosine Integral $\int_{0}^{\frac{\pi}{2}} \frac{ \sin(kx) \ +\ \cos(kx)}{\sin x \ +\ \cos x} \ dx$ What is the value of the integral
\begin{align}
I(k) = \int_{0}^{\frac{\pi}{2}} \frac{ \sin(kx) + \cos(kx)}{\sin(x) + \cos(x)} \ dx
\end{align}
where $k$ is an integer ?
Is it possible to evaluate a related integra... | Depending on even or odd $k$, the integrals
\begin{align}
I_k = \int_{0}^{\frac{\pi}{2}} \frac{ \sin(kx) + \cos(kx)}{\sin x + \cos x} \ dx
\end{align}
exhibit different behaviors. Their close-forms are derived separately below
\begin{align}
I_{2m+1}
=& \>2\cos\frac{m\pi}2\int_0^{\frac\pi4}\frac{\cos(2m+1)x}{\cos x}dx
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/839131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$? I have no idea how to start, it looks like integration by parts won't work.
$$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$
If someone could shed some light on this I'd be very thankful.
| Note
\begin{eqnarray}
I&=&\int_0^\infty \frac{x[(x^2+1)-1]^2\sin x}{(1+x^2)^3}dx\\
&=&\int_0^\infty \frac{x\sin x}{1+x^2}dx-2\int_0^\infty \frac{x\sin x}{(1+x^2)^2}dx+\int_0^\infty \frac{x\sin x}{(1+x^2)^3}dx.
\end{eqnarray}
From
$$ \int_0^\infty \frac{\cos(ax)}{b^2+x^2}dx=\frac{\pi}{2b}e^{-ab}, a>0, b>0, $$
we have
\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/839426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 4
} |
How can I plot the set $M:=\left\{\left(\begin{array}{c}x\\y\end{array}\right)\in \mathbb R^2:9x^4-16x^2y^2+9y^4\leq9\right\}$ I need to plot the following set:
$$M:=\left\{\left(\begin{array}{c}x\\y\end{array}\right)\in \mathbb R^2:9x^4-16x^2y^2+9y^4\leq9\right\}$$
I have solved the equation $9x^4-16x^2y^2+9y^4-9=0$ f... | Note that in polar co-ordinates $x=r\cos \theta, y=r\sin \theta$ the function becomes $$r^4(9\cos^4\theta-16\cos^2\theta\sin^2\theta+9\sin^4\theta)$$
Then $\cos^4\theta+\sin^4\theta =(\cos^2\theta+\sin^2\theta)^2-2\cos^2\theta\sin^2\theta$ so the trigonometric part becomes $$9-34\cos^2\theta\sin^2\theta=9-\frac {17}2\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Does the series $\sum_{n=1}^\infty\frac{\sin n}{\ln n+\cos n}$ converge? $$\sum_{n=1}^\infty\frac{\sin n}{\ln n+\cos n}$$
My guess is "yes", but I can't prove it.
| This method was inspired by the OP's heuristic argument in a comment to the question. We approximate the summand by Taylor polynomials, but with more and more terms as $n$ grows.
We need the fact that for every integer $k\ge1$,
$$
\bigg| \sum_{B\le n< C} \sin n \cdot \cos^k n \bigg| = O(k^7)
$$
uniformly for integers $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/841080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 4,
"answer_id": 1
} |
$\left(\sqrt{8}+\sqrt{2}\right)^2$ = 18 why?? I would like to now why
$$\left(\sqrt{8}+\sqrt{2}\right)^2$$
is equal to $18$? Please provide me
the proccess. Thank you.
| It does look counter-intuitive, doesn't it? But if you step through the multiplication, I think it will become crystal clear to you. First of all, $$(\sqrt{8} + \sqrt{2})^2 = (\sqrt{8} + \sqrt{2})(\sqrt{8} + \sqrt{2})$$, right? Applying FOIL (First, Outer, Inner, Last), we get $$\sqrt{8} \sqrt{8} = 8$$ $$\sqrt{8} \sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 5
} |
Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$ Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$
I went wrong somewhere, this is what I have so far (this is in polar):
$z=4\left(\cos\left(\frac{11\pi}{6}\right)+\sin\left(\frac{11\pi}{6}\right)\right) $
$w=\sqrt2\left(\cos\left(\frac{7\p... | You have the angle wrong for $w$. It should be $3\pi/4$, right?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/845086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Integer sum as binomial coefficient What's the rule for expressing integer sums as binomial coefficients? That is, for $p=1$ it is $$\sum_{n=1}^N n^p = {{N+1}\choose 2} $$
What is it for higher powers?
| $\def\str#1#2{\left\{#1\atop#2\right\}}$
Let $\str{n}{m}$ be the Stirling number of the second kind, It can be defined by the formula
$$
X^p=\sum_{m=0}^p\str{p}{m}X(X-1)\cdots(X-m+1)\tag 1
$$
This implies that
$$\eqalign{
n^p&=\sum_{m=0}^p m!\str{p}{m}\binom{n}{m}\tag 2\cr
&=\sum_{m=0}^p m!\str{p}{m}\left(\binom{n+1}{m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/846144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Algebraic manipulation of floors and ceilings I am trying to solve the summation
$$
\sum_{n=3}^{\infty} \frac{1}{2^n} \sum_{i=3}^{n} \left\lceil\frac{i-2}{2}\right\rceil
$$
I will list some of the simplifications that I've found so far, and then get around to asking my question. Please feel free to point out any logica... | \begin{align*}S&=\sum_{n=3}^\infty\frac{1}{2^n}\sum_{i=3}^n\left\lceil\frac{i-2}{2}\right\rceil&=\sum_{n=3}^\infty\frac{1}{2^n}\sum_{i=3}^n\left\lfloor\frac{i-1}{2}\right\rfloor\\ &=\sum_{n=3}^\infty\frac{1}{2^n}\sum_{i=2}^{n-1}\left\lfloor\frac{i}{2}\right\rfloor &=\sum_{n=3}^\infty\frac{1}{2^n}\left\lfloor\frac{(n-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/847183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Finding closed form for $1^3+3^3+5^3+...+n^3$ I'd like to find a closed form for $1^3+3^3+5^3+...+n^3$ where $n$ is an odd number.
How would I go about doing this?
I am aware that $1^3+2^3+3^3+4^3+...+n^3=\frac{n^2(n+1)^2}{4}$ but I'm not too sure how to proceed from here.
My gut feeling is telling me to multiply the ... | By factorizing $2^3 = 8$ from the even terms, we can re-express the sum of the even terms:
$$2^3 + 4^3 + \dots + (n-1)^3 = 8\left(1^3 + 2^3 + \dots + \left(\frac{n-1}{2}\right)^3\right)$$
Simply subtract this from $1^3 + 2^3 + \dots + n^3$ to give the sum of the remaining odd terms. You can compute the formula for the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/848087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 5
} |
limits using $ \epsilon - \delta $ to prove two variable function I'm trying to use the $ \epsilon - \delta $ argument to prove $\lim_{(x,y) \rightarrow (1.1)} \frac{2xy}{x^2+y^2} =1$. I know that I need to show that $\forall \epsilon>0, \exists \delta>0$ s.t. for all (x,y) in the domain of f, $| \frac{2xy}{x^2+y^... | So you have
$$
\frac{(x-y)^2}{x^2+y^2}.
$$
Since you can always assume that $x> 1/2$ and $y>1/2$,
$$
x^2+y^2 > \frac{1}{2},
$$
and
$$
\frac{(x-y)^2}{x^2+y^2} < 2(x-y)^2.
$$
Now, $(x-y)^2 = |(x-y)(x-y)|< (|x|+|y|)|x-y| < \left(\frac{3}{2}+\frac{3}{2} \right) |x-y|$ because you can also assume $x<3/2$, $y<3/2$. I leave ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/848350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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A simple conditional probability problem Assume that two fair dice are rolled one at a time. Given that the sum of the two numbers that occured was at least $7$, compute the probability that it was equal to $7$.
I tried computing the probability as follows:
$$P \left( X+Y=7| X+Y \geq 7 \right)=\frac{P \left(X+Y=7 \righ... | $P(X+Y=7) = 1/6$
By Symmetry
$P(X+Y>7) = P(X+Y<7) = (1 - 1/6)/2$
$P(X+Y>=7) = P(X+Y>7)+P(X+Y=7) = 5/12 + 1/6$
$P( X+Y=7| X+Y >= 7) = \frac{1/6}{5/12+1/6} = 6/21$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/848638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Two diophantine equations with lots of unknowns Is it possible (tractable) to determine if the following system of equations has any nontrivial solutions (ie, none of the unknowns are zero) in the domain of integers?
$$A^2 + B^2=C^2 D^2$$
$$2 C^4 + 2 D^4 = E^2 + F^2$$
| To solve,
$$A^2+B^2=C^2 D^2\\
(2C)^4+(2D)^4=E^2+F^2$$
Choose,
$$\begin{aligned}
A&=2(ac-bd)(ad+bc)\\
B&=(ac-bd)^2-(ad+bc)^2\\
C&=a^2+b^2\\
D&=c^2+d^2\\
E&=(a^2+b^2 )^2-(c^2+d^2 )^2\\
F&=(a^2+b^2 )^2+(c^2+d^2 )^2\\
\end{aligned}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/850631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$f(x,y)=(x^2-y^2,2xy)$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$ Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation
$$f(x,y)=(x^2-y^2,2xy).$$
Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$.
What is the set $f(A)?$
| The inverse function theorem does not seem to help since we are talking about global invertability. To show injective assume that
$$x^2-y^2=w^2-z^2$$ and
$$2xy=2wz$$
We then have on squaring and adding,
$$(x^2+y^2)^2=(w^2+z^2)^2$$ this implies that
$$x^2+y^2=w^2+z^2$$ from which we get $x^2=w^2$ and since we assume t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Producing a CDF from a given PDF So I have this PDF:
$$
f(x)=
\begin{cases}
x + 3 & \text{ for } -3 \leq x < -2\\
3 - x & \text{ for } 2 \leq x < 3\\
0 & \text{ otherwise}
\end{cases}
$$
To make this a CDF, I have integrated the PDF from $-\infty$ to some value, $x$.
$$
F(x)=
\begin{cases}
\frac{x^2}{2} + 3x + \fra... | Hint: integrate $ 3-x$. Say, integral is $f(x)$. Plug in $f(3)=1$ to get the value of constant
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/851854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
} |
Equation of a line tangent to a circle: why it works? I've the problem following: "Determine the points on the circumference of the circle $x^2 + y^2 = 1$ whose tangent lines are
and the point $(3,0)$." I found the derivative $\dfrac{\mathrm{d} y}{\mathrm{d} x}=-\dfrac{x}{y}$ and I managed to solve the exercise by doi... | We do it the calculus way (there is a much simpler geometric way).
Let the point(s) of tangency be $(a,b)$. Then by your calculation, the slope of the tangent line is $-\frac{a}{b}$.
The tangent line, we are told, goes through the external point $(3,0)$. The line joining $(a,b)$ to $(3,0)$ has slope $\frac{b}{a-3}$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/852911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is the value of $\lfloor{100N}\rfloor$ What is the value of $\lfloor{100N}\rfloor$ where $\displaystyle N= \lim\limits_{a\,\rightarrow\,\infty}\sum\limits_{x=-a}^{a}\frac{\sin x}{x}$.
This is a part of a bigger problem that I was solving. I need the exact integer value of $\lfloor{100N}\rfloor$ .
I was only able t... | Note
\begin{eqnarray*}
\sum_{k=1}^n\cos(kx)=-\frac{1}{2}+\frac{\sin(\frac{n}{2}+1)x}{2\sin\frac{x}{2}}
\end{eqnarray*}
and hence
\begin{eqnarray*}
\sum_{k=1}^n\frac{\sin k}{k}&=&\int_0^1\sum_{k=1}^n\cos(kx)dx\\
&=&-\frac{1}{2}+\int_0^1\frac{\sin(\frac{n}{2}+1)x}{2\sin\frac{x}{2}}dx\\
\end{eqnarray*}
Using the followin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/854248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates I'm having a problem converting $\int\limits_1^2 \int\limits_0^ \sqrt{2x-x^2} \frac{1}{(x^2+y^2)^2} dy dx $ to polar coordinates.
I drew the graph using my calculator, which looked like half a circle on the x axis.
I kno... | $\int_{1}^{2} \int_{0}^{\sqrt{2 x-x^{2}}} \frac{1}{\left(x^{2}+y^{2}\right)^{2}} d y d x$ is an integration over region $y=0$ to $y=\sqrt{2 x-x^{2}}$ and $x=1$ to $x=2$ as shaded region in below graph. Now we will change $xy$ to polar co-ordinate. Using $x=r \cos \theta$, $y=r \sin \theta$ and $dxdy=r dr d\theta$. Aft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/855362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Orthogonal Projection of $v$ on sub-space $U$
What is the orthogonal projection of $v = (1,5,-10)$ on the sub-space $U = Sp\{(5,-2,1),(1,2,-1)\}$?
Well, I managed to compute it on the way I've been taught by the book, but it doesn't seem to work. Any chance you guys can solve it? If needed, I can provide the way I tr... | Another approach is to compute the normal to the subspace using the cross product:
$$
\begin{pmatrix}5,-2,1\end{pmatrix}\times\begin{pmatrix}1,2,-1\end{pmatrix}=\begin{pmatrix}0,6,12\end{pmatrix}
$$
Then subtract the component of $\begin{pmatrix}1,5,-10\end{pmatrix}$ in that direction:
$$
\begin{align}
&\begin{pmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/856246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to show that $\lim\limits_{n\to\infty}n^{2/3}a_{n}=\sqrt[3]{2}/\Gamma{(1/3})$ Let $$\left(\dfrac{1+x}{1-x}\right)^{1/3}=\sum_{n=0}^{\infty}a_{n}x^n,|x|<1$$
Show that
$$\lim_{n\to\infty}n^{2/3}a_{n}=\dfrac{\sqrt[3]{2}}{\Gamma{\left(\dfrac{1}{3}\right)}}$$
| Write
$$
\left(\dfrac{1+x}{1-x}\right)^{1/3} = \frac{(1+x)^{1/3} - 2^{1/3}}{(1-x)^{1/3}} + \frac{2^{1/3}}{(1-x)^{1/3}} =: f(x)+g(x).
$$
The function $f$ is continuous on the disk $|z| \leq 1$, so by Darboux's method (Thm. VI.14 in Analytic Combinatorics) we have
$$
[x^n] f(x) = o\left(\frac{1}{n}\right).
$$
The coeffic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/856784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Infinite product: $\prod_{k=2}^{n}\frac{k^3-1}{k^3+1}$ I am trying to find $$\lim_{n\rightarrow \infty}\prod_{k=2}^{n}\frac{k^3-1}{k^3+1}.$$I have no idea how I can start. Please help me.
Thanks!
| The main idea is to use a telescope product of the form $ \prod \frac{b_j}{b_{j+2}} \frac{a_{j+1}}{a_j}$. We have$$ \lim_{n \to \infty} \prod_{k=2}^n \frac{k^3-1}{k^3+1}=\lim_{n \to \infty}\prod_{k=2}^n \frac{k-1}{k+1} \frac{k^2+k+1}{k^2-k+1}=\lim_{n \to \infty}\prod_{k=2}^n \frac{k-1}{(k+2)-1} \frac{(k+1)^2-(k+1)+1}{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/856965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How does $n+2\mid(n^2+n+1)(n^2+n+2)$ imply $n+2\mid12$? I'm trying to show that if $n+2\mid(n^2+n+1)(n^2+n+2)$, then $n+2 \mid 12$ for all $n\in \mathbb{N}$.
| Hint
$$(n^2+n+1)(n^2+n+2)=n^4+2n^3+4n^2+3n+2$$
and
$$ n^4+2n^3+4n^2+3n+2=(n+2)(n^3+4n-5)+12$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/857798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
closed form for $\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$ $$\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$$
my friend posted the integral on the fb and I tried to solve it but I faild because I have little information about bessel function
so could some one help ?
| Using the series expansion for the Bessel function the integral is as follows:
\begin{align}
I &= \int_{0}^{1} x^{a+1} (1-x^{2})^{b} J_{a}(cx) \ dx \\
&= \sum_{n=0}^{\infty} \frac{(-1)^{n} (c/2)^{2n+a}}{n!(n+a)!} \ \int_{0}^{1} x^{2n+2a+1} (1-x^{2})^{b} \ dx.
\end{align}
Making the substitution $x = \sqrt{t}$ the integ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/859625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Compute integral: $\int_{0}^{\pi/2}\log(a^2\sin^2 x+b^2\cos^2 x )dx$ This is a integral for a calculus exam, and I have no idea how to solve it.
$$\int_0^{\frac{\pi}{2}} \log \big( a^2 \sin^{2}(x)+b^2 \cos^{2}(x) \big) \, \mathrm{d}x$$
| Consider
$$I(b)=\int_0^{\pi/2} \ln\left(a^2\sin^2x+b^2\cos^2x\right)\,dx $$
$\displaystyle
\begin{aligned}
\Rightarrow \frac{\partial I}{\partial b} &=2b\int_0^{\pi/2} \frac{\cos^2x}{a^2\sin^2x+b^2\cos^2x}\,dx\\
&=2b\int_0^{\pi/2} \frac{dx}{a^2\tan^2x+b^2}\\
&=2b\int_0^{\infty} \frac{dt}{(a^2t^2+b^2)(1+t^2)}\,\,\,\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Double Angle Trigonometry Question So there is this question which consists of 2 parts.
$$
a) \text{ Simplify } \frac{\sin2x}{1+\cos2x} \\
b) \text{ Hence, find the exact value of tan 15.}
$$
So far I've discovered that $ \text{a)} \tan x $
But I have no idea how to begin on part $b$, although I'm guessing the answer's... | We have: $\dfrac{\sin(2x)}{1+\cos(2x)}$
$=\hspace{12 mm}\dfrac{2\sin(x)\cos(x)}{1+2\cos^{2}(x)-1}$
$=\hspace{12 mm}\dfrac{2\sin(x)\cos(x)}{2\cos^{2}(x)}$
$=\hspace{12 mm}\dfrac{\sin(x)}{\cos(x)}$
$=\hspace{12 mm}\tan(x)$
Then, we want to evaluate $\tan(15)$.
We can do this using the original expression $\dfrac{\sin(2x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/861024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Solve the System of Equations in Real $x$,$y$ and $z$ Solve for $x$,$y$ and $z$ $\in $ $\mathbb{R}$ if
$$\begin{align} x^2+x-1=y \\
y^2+y-1=z\\
z^2+z-1=x \end{align}$$
My Try:
if $x=y=z$ then the two triplets $(1,1,1)$ and $(-1,-1,-1)$ are the Solutions.
if $x \ne y \ne z$ Then we have
$$\begin{alig... | $\textbf{Hint:}$Note that $\displaystyle x^2-x-1=y$ is equal $\displaystyle (x-\frac{1}{2})^2-\frac{5}{2}=y-\frac{1}{2}$, so if you substitute $x_1=x-\frac{1}{2}$, $y_1=y-\frac{1}{2}$ and $z_1=z-\frac{1}{2}$ you get:
$$x_1^2-\frac{5}{2}=y_1$$
$$y_1^2-\frac{5}{2}=z_1$$
$$z_1^2-\frac{5}{2}=x_1$$
Substitute (1) to (2), ne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Calculus area between graph and x axis
How to solve this question ?
The integral would be 0.3x^3+xc^2 + c
0.3(64)^3 +(64)c^2 +c =0
c= 78643.2/( -64+c)
am I right ?
| The problem should be restated because the area "between" the function $y=x^2-c^2$ and $y=0$ is always equal to $\infty$.
I would restated the problem as follows
Find the exact positive value of $c$ if the area of the region bounded by $y=x^2-c^2$ and the x-axis is 64.
Next, make some sketches.
Clearly the region bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$? I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of integration by parts. However, I don't think that's th... | As noted in the comments, the integral is:
$$I=\int_{-\infty}^{\infty} \frac{x^2}{x^4+x^2+1}\,dx=2\int_0^{\infty} \frac{x^2}{x^4+x^2+1}\,dx\,\,\,(*)$$
With the change of variables $x\mapsto 1/x$,
$$I=2\int_{0}^{\infty} \frac{1}{x^4+x^2+1}\,dx\,\,\,\,\,\,(**)$$
Add $(*)$ and $(**)$ i.e
$$2I=2\int_0^{\infty} \frac{1+x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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How can we apply the forward Euler method to $x''=x^2$? If we want to apply the forward Euler method to $x''=x$ with $x(0)=0, x'(0)=1$, we can introduce a new function $$u:=\begin{bmatrix}x'\\x \end{bmatrix}$$
then $$u'=\begin{bmatrix}0&1\\1&0\\ \end{bmatrix}u \ \ \text{and} \ \ u(0)=\begin{bmatrix}1\\0 \end{bmatrix}.$... | Let $u = \begin{bmatrix} x \\ x'\end{bmatrix}$. Then, $u' = \begin{bmatrix} x' \\ x''\end{bmatrix} = \begin{bmatrix} x' \\ x^2\end{bmatrix}$ with $u(0) = \begin{bmatrix} 0 \\ 1\end{bmatrix}$.
So, taking forward Euler steps yields:
$u(h) \approx u(0) + hu'(0) = \begin{bmatrix} 0 \\ 1\end{bmatrix} + h\begin{bmatrix} 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/868522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is $\int^x \cos \frac1t$ differentiable at zero? From Spivak's Calculus, 4th ed., exc 14-20:
Let $$f(x) = \begin{cases} \cos \frac1x, & x\neq 0\\ 0, &x=0.
\end{cases}$$ Is the function $\int_0^xf$ differentiable at zero?
I'm having trouble with this one; I'm sure I'm just missing something easy.
We seek
$$\lim_{... | Without that substitution, we can note that
$$g(x) = \begin{cases} -x^2\sin \frac{1}{x} &, x \neq 0 \\ \quad 0 &, x = 0\end{cases}$$
is differentiable everywhere, with $g'(0) = 0$ and
$$g'(x) = \cos \frac{1}{x} -2x\sin\frac{1}{x}$$
for $x\neq 0$. So
$$\int_0^h \cos \frac{1}{\xi}\,d\xi = \int_0^h g'(\xi) + 2\xi\sin \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Evaluate expression in the form $a+bi$. So, I have to evaluate $\sqrt{-3}\sqrt{-12}$ into the form $a+bi$.
I know that $i^2 = -1$ so $i = \sqrt{-1}$
What I have done is:
$$\begin{align}\sqrt{-3}\sqrt{-12}
&= \sqrt{3(-1)}\sqrt{12(-1)}\\
&= \sqrt{3}i\sqrt{12}i\\
&= \sqrt{3}\cdot i\cdot 2\sqrt{3}\cdot i\\
&= 3\cdot 2\... | Yes it's correct. Just take $a = -6$ and $b = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/872103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How find this integral $I=\int_{0}^{\frac{\pi}{2}}(\ln{(1+\tan^4{x})})^2\frac{2\cos^2{x}}{2-(\sin{(2x)})^2}dx$
Find the value:
$$I=\int_{0}^{\frac{\pi}{2}}(\ln{(1+\tan^4{x})})^2\dfrac{2\cos^2{x}}{2-(\sin{(2x)})^2}dx$$
I use computer have this reslut
$$I=-\dfrac{4\pi}{\sqrt{2}}C+\dfrac{13\pi^3}{24\sqrt{2}}+\dfrac{9}{... | After performing the initial arctangent substitution, the resultant integral may be rewritten as the second derivative of a beta function and evaluated accordingly:
$$\begin{align}
I
&=\int_{0}^{\frac{\pi}{2}}\frac{2\cos^2{\theta}}{2-\sin^2{(2\theta)}}\log^2{\left(1+\tan^4{\theta}\right)}\,\mathrm{d}\theta\\
&=\int_{0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
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Proving $\frac{p-1}{2}!\equiv (-1)^t$ where $t$ is the number of integers which are not quadratic squares
Prove that $\frac{p-1}{2}!\equiv (-1)^t$ where $t$ is the number of integers $0<a<\frac{p}{2}$ which are not quadratic squares $\pmod p$ ($p\equiv3\bmod4$)
I don't know really from where to start (we know from w... | $$\left ( \left ( \frac{p-1}{2} \right )! \right )^2= 1^2 \cdot 2^2 \cdot 3^2 \cdots \left (\frac{p-1}{2} \right )^2=1 \cdot 2 \cdot 3 \cdots \frac{p-1}{2} \cdot 1 \cdot 2 \cdot 3 \cdots \frac{p-1}{2} \\\equiv 1 \cdot 2 \cdot 3 \cdots \frac{p-1}{2} \cdot (-1) \cdot (p-1) \cdot (-1) \cdot (p-2) \cdots (-1) \cdot \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/874793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Finding $\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $ Help me to simplify:$$\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $$
I got a hunch that it will depend on whether $n$ is a multiple of $6$ and equals to $\frac{2^n+2}{3}$ when $n$ is a multiple of $6$.
| If $1+\zeta+\zeta^2=0$, try to compute $(1+\zeta^0)^n + (1+\zeta)^n + (1+\zeta^2)^n$. Hint: $\zeta^3=1$, and, if $k$ is not divisible by $3$, $1+\zeta^k+\zeta^{2k}=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/875153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 2
} |
If $a^2=b^2+c^2$ and $0If $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, prove
(a) if $n>2$ then $a^n>b^n+c^n$,
(b) if $0<n<2$ then $a^n<b^n+c^n$.
Part (a) was easy to prove: $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, so $a>b$ and $a>c$. Then O can show: $b^n+c^n=b^2(b^{n-2})+c^2(c^{n-2})<b^2(a^{n-2})+c... | These may help you:
First prove for $n = 1$ (Easy, as $a = \sqrt{b^2+c^2} < \sqrt{b^2+c^2 + 2bc} = b +c$).
Then, separately try for $n<1$ and $1<n\leq2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/875549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Proving Fibonacci inequality I didn't see a question regarding this particular inequality, but I think that I have shown by induction that, for $n>1$. I am hoping someone can verify this proof.
$$\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}<F_{n+1}<\left(\frac{1+\sqrt{5}}{2}\right)^{n}$$
Here we state that $F_0=0, F_1=1$. ... | Your proof is valid, but i like a little less tedious proof:
Note that $$\phi^2=\phi+1$$
$$\phi^3=\phi^2+\phi=2\phi+1$$
$$\phi^4=2\phi^2+\phi=3\phi+2$$
$$\phi^5=3\phi^2+2\phi=5\phi+3$$
see the Fibonacci numbers in the coeficcients, now generally:
$$\phi^n=F_n\phi+F_{n-1}$$, so we can rewrite your problem:
$$\phi^{n-1}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/876002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Which one is correct for $\sqrt{-16} \times \sqrt{-1}$? $4$ or $-4$? As we can find in order to evaluate $\sqrt{-16} \times \sqrt{-1}$, we can do it in two ways.
FIRST
\begin{align*}
\sqrt{-16} \times \sqrt{-1} &= \sqrt{(-16) \times (-1)}\\
&= \sqrt{16}\\
&=4
\end{align*}
SECOND
\begin{align*}
\sqr... | $$\sqrt a\cdot\sqrt b=\sqrt{ab}$$ only work if $a,b\ge0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/876084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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} |
Find $\tan x $ if $\sin x+\cos x=\frac12$ It is given that $0 < x < 180^\circ$ and $\sin x+\cos x=\frac12$, Find $\tan x $.
I tried all identities I know but I have no idea how to proceed. Any help would be appreciated.
| Or: $1 + 2\sin{x }\cos{x} = \frac{1}{4},$ so $\sin{x}\cos{x} = \frac{-3}{8},$ leading to $\sin{x} - \cos{x} = \pm \sqrt{ \frac{1}{4} + \frac{3}{2}} = \pm \frac{\sqrt{7}}{2},$ allowing you to solve for $\sin{x}$ and $\cos{x}$ (making use of the fact that $0 < x < \pi$ (radians) so that $\sin{x} >0$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/877585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 7
} |
Finding common ratio from two sums I'm struggling with this very basic question on the binomial theorem:
The sum of the first and second terms of a geometric progression is 12, and the sum of the third and fourth term is 48. Find the two possible values of the common ratio and the corresponding values of the first ter... | From your first two equations we get
$$\frac{48}{12}=\frac{ar^2+ar^3}{a+ar}=r^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/880751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.