Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Calculus Question - change order of integration Can We always change the order of integration in double integrals ?
| Here is a counterexample that shows the value might not coincide.
Consider the function
$$
\frac{x^2-y^2}{(x^2+y^2)^2}
$$
A $y$-primitive for this on $[1,\infty)$ is
$$
\frac{y}{x^2+y^2}
$$
which simplifies to $$-\frac{1}{1+x^2},$$ when evaluated from $1$ to $\infty$.
Knowing this, and the fact that $$\int \frac{1}{1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/244092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Function expansion help I want to write a function, $f(k,a,b)$, I made, in terms of combinations of the fractional part function, $$ j\left\{\frac{c \ }{d}k\right\},$$ where $c,d,$ and $j$ are any integers.
The function is as follows: $f(k,a,b)=1$ if $k\equiv b$ mod a
and $f(k,a,b)=0$, if it is not
I need a general m... | Let C be a (m-1)*(m-1) matrix of numbers $\large\ c_{ij}=\left\{\Large\frac{i\cdot j}{m}\right\}$
Let $D=C^{-1}$
Then
$\large f(k,m,1)=\sum_{i=1}^{m-1}d_{i1}\Large\{\frac{i}{m}\cdot\large k\}$
$\large f(k,m,r)=f(k-r+1,m,1)$
Example: $\ \ \ \ \ m=6$
$$
C=\frac{1}{6}\cdot\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
2 & 4 & 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/244648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why are these two expressions different in this induction problem? Prove with $n \ge 1$:
$$\frac{3}{1\cdot2\cdot2} + \frac{4}{2\cdot3\cdot4}+\cdots+\frac{n+2}{n(n+1)2^n} = 1 - \frac{1}{(n+1)2^n}$$
First, I prove it for $n=1$:
$$\left(\frac{1+2}{1(1+1)2^1} = 1-\frac{1}{(1+1)2^1}\right) \implies \left(\frac{3}{4} = 1- \f... | Elevating comment to answer, at suggestion of OP:
In the last term of the last displayed equation, there is a $5+1$ where there should be a $5+2$. Jonas Meyer notes that this correction gets rid of the discrepancy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/245505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Integral by partial fractions
$$ \int \frac{5x}{\left(x-5\right)^2}\,\mathrm{d}x$$ find the value of the constant when the antiderivative passes threw (6,0)
factor out the 5, and use partial fraction
$$ 5 \left[\int \frac{A}{x-5} + \frac{B}{\left(x-5\right)^2}\, \mathrm{d}x \right] $$
Solve for $A$ and $B$.
$A\left... | Your solution is correct, but books solution is also. Differentiate the solutions and you will see, that both of them are Antiderivatives.
Moreover it is:
$$ \frac{5}{x-5} \left(\left(x-5\right) \ln \vert x - 5 \vert - x \right)
= 5 \left(\ln \vert x - 5 \vert - \frac{x-5+5}{x-5}\right) = 5 \left(\ln \vert x - 5 \vert... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Area of an ellipse An ellipse has equation :
$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0$$
Can you provide an optimum method to find it's area?
| When rotating conics in implicit form
$$
Ax^2+Bxy+Cy^2+Dx+Ey+F=0\tag{1}
$$
around the origin there are 5 invariants:
$$
\begin{array}{rl}
I_1&=A+C\\
I_2&=(A-C)^2+B^2\\
I_3&=D^2+E^2\\
I_4&=(A-C)(D^2-E^2)+2DEB\\
I_5&=F\tag{2}
\end{array}
$$
Assuming that we have rotated to eliminate $B$, we have
$$
Ax^2+Cy^2+Dx+Ey+F=0\ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 0
} |
Find parametrization of a manifold Find parametrization of $x^2y^2+y^2z^2+z^2x^2=xyz$, $x,y,z>0$.
I've checked it's manifold but I can't find parametrization(s).
| So, $$\frac {xy}z+\frac{yz}x+\frac{zx}y=1$$
As $x,y,z>0$
we can write $\frac {xy}z+\frac {yz}x=\cos^2\theta$ and $\frac{zx}y=\sin^2\theta$
So, $\frac {xy}{z\cos^2\theta}+\frac {yz}{x\cos^2\theta}=1$
So, we can write $\frac {xy}{z\cos^2\theta}=\cos^2\phi$ and $\frac {yz}{x\cos^2\theta}=\sin^2\phi$ as $\frac {xy}{z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/248603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find Jordan-Normal form of a given matrix I am trying to write the matrix $A=\begin{pmatrix} 2 & 2 & -5 \\ 3 & 7 & -15 \\ 1 & 2 & -4 \end{pmatrix}$ in Jordan-Normal Form.
It's characteristic polynomial is $|A-xI|=x^3-5x^2+7x-3=(x-1)^2(x-3)=0$
So, it has eigenvalues 1 and 3. I calculated three eigenvectors $v_1=\left(\... | Everything you have calculated is correct. A Jordan-Normal-Form for your matrix is $$\pmatrix{3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1}$$
Another solution would be for example $$\pmatrix{1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3}$$
The Jordan-Normal-Form do not have to be unique and depends on how you arange the vectors in your m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/249948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to simplify $\frac{4 + 2\sqrt6}{\sqrt{5 + 2\sqrt{6}}}$? I was tackling through an olympiad practice book when I saw one of these problems:
If $x = 5 + 2\sqrt6$, evaluate $\Large{x \ - \ 1 \over\sqrt{x}}$?
The answer written is $2\sqrt2$, but I can't figure my way out through the manipulations. I just know that ... | $$\frac{x-2}{\sqrt x}=\frac{4+ 2\sqrt6}{\sqrt{5 + 2\sqrt6}}=\frac{4+ 2\sqrt6}{\sqrt{5 + 2\sqrt6}}\cdot\frac{\sqrt{5-2\sqrt 6}}{\sqrt{5-2\sqrt 6}}=\left(4+2\sqrt 6\right)\sqrt{5-2\sqrt 6}\Longrightarrow$$
$$(x-2)^2=x(40+16\sqrt 6)(5-2\sqrt 6)=x(8)\Longrightarrow $$
$$x^2-12x+4=0\ldots...$$
Take it from here
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/249993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 4
} |
Computation of a certain integral I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do for this one too.
$$\int_{0}^{2\pi} \frac{1}{a-\cos(x)}dx, \text{ with } a > 1.$$
Any ... | $$\cos x = \frac{e^{i x}+e^{-ix}}{2}$$
Thus
$$\int_0^{2\pi} \frac{dx}{a-\frac{e^{i x}+e^{-ix}}{2}} = \oint_C \frac{1}{a-\frac{z+z^{-1}}{2}} \frac{dz}{iz} = - i\oint_C \frac{dz}{az-\frac{z^2+1}{2}} = 2i\oint_C \frac{dz}{z^2-2az+1}$$
where $C$ describes the unit circle $|z|=1$, centred at the origin, parametrized by $e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/250250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Problem when integrating $e^x / x$. I made up some integrals to do for fun, and I had a real problem with this one. I've since found out that there's no solution in terms of elementary functions, but when I attempt to integrate it, I end up with infinite values. Could somebody point out where I go wrong?
So, I'm trying... | This part looks right:
$$\int{\frac{e^x}{x}} \, dx = \frac{e^x}{x} + \frac{e^x}{x^2} + \frac{2e^x}{x^3} + \frac{6 e^x}{x^4} + \frac{24 e^x}{x^5} + \cdots+ \frac{n!e^x}{x^{n+1}}+(n+1)!\int \frac{e^x}{x^{n+1}}$$
When you say "repeating to infinity" you want to take the limit of that...in order for your equality to hold, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/251795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 3,
"answer_id": 1
} |
Inequality. $(a^2+bc)(b^2+ca)(c^2+ab) \geq abc(a+b)(b+c)(c+a)$ Let $a,b,c$ be three real positive(strictly) numbers. Prove that:
$$(a^2+bc)(b^2+ca)(c^2+ab) \geq abc(a+b)(b+c)(c+a).$$
I tried :
$$abc\left(a+\frac{bc}{a}\right)\left(b+\frac{ca}{b}\right)\left(c+\frac{ab}{c}\right)\geq abc(a+b)(b+c)(c+a) $$
and now I wa... | $$LHS-RHS=abc\sum_{sym} a(a-b)(a-c)+\frac{1}{2}(ab+bc+ca)\left[c^2(a-b)^2+a^2(b-c)^2+b^2(c-a)^2\right]\ge 0$$
True by Schur 3 deg
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/253015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 6
} |
How to find these integrals? How to find the following two integrals?
$$\int_{0}^{1}{\sqrt{{{x}^{3}}-{{x}^{4}}}dx}$$
and
$$\int_{0}^{1}{x\sqrt{{{x}^{3}}-{{x}^{4}}}dx}$$
| $$I:=\int_0^1 x\sqrt {x^3-x^4}\,dx=\int_0^1 x^2\sqrt{x-x^2}\,dx=\int_0^1x^2\sqrt{\frac{1}{4}-\left(\frac{1}{2}-x\right)^2}\,dx=$$
$$=\frac{1}{2}\int_0^1x^2\sqrt{1-\left(1-2x\right)^2}\,dx $$
Substitute now
$$\sin u=1-2x\Longrightarrow \cos u\,du=-2\,dx\,,\,x=0\Longrightarrow u=\frac{\pi}{2}\;\;,\;x=1\Longrightarrow u=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving that the following series is convergent: $\sum\limits_{i=1}^\infty \left({n^2+1\over n^2+n+1}\right)^{n^2}$ Can someone please help me prove that this series is convergent?
$$ \sum_{i=1}^\infty \left({n^2+1\over n^2+n+1}\right)^{n^2} $$
I guess I'm supposed to show that the limit of the sequence is an "e" limit... | Note that $${\left( \dfrac{n^2+n+1}{n^2+1}\right)^{n^2}}={\left(1+ \dfrac{n}{n^2+1}\right)^{\frac{n^2+1}{n}\cdot\frac{n^3}{n^2+1}}}\geqslant 2^{\frac{n}{2}},$$ because $2<\left(1+ \dfrac{n}{n^2+1}\right)^{\frac{n^2+1}{n}}<3$ and $\frac{n^3}{n^2+1}\geqslant \frac{n}{2}$ for $n\geqslant 1.$ Therefore,
$$\left( \dfrac{n^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/254853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Number of ordered sets of integers How many ordered sets of integers $(x,y,z)$ satisfying $$x,y,z \in [-10,10]$$ are solutions to the following system of equations:
$$x^2y^2+y^2z^2=5xyz$$
$$y^2z^2+z^2x^2=17xyz$$
$$z^2x^2+x^2y^2=20xyz$$
By... | If any two of $x,y,z$ are $0$, all the equations say $0 = 0$.
On the other hand, if only one is $0$, it's easy to see that another must also be $0$.
So now let's assume none of them are $0$.
Then we can cancel a $y$, $z$, or $x$ respectively from each equation, leaving
$$ \eqalign{(x^2 + z^2) y &= 5 x z\cr
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How do you integrate$\int_{-\infty}^{\infty} dw \frac{1}{(\alpha^2-w^2)^2+ w^2\beta^2} $ I am having trouble integrating the following expression appearing in a mechanical problem:
$$\int_{-\infty}^{\infty} dw \frac{1}{(\alpha^2-w^2)^2+ w^2\beta^2} $$
I tried using the residue theorem, but having a polynom of degree 4 ... | Here is how we might go about it while minimizing the use of computer algebra tools. The idea is to compute the integral on a contour consisting of the line segment from $-R$ to $R$ on the real axis and a semicircle of radius $R$ in the upper half plane, letting $R$ go to infinity. Since our function $$f(w) = \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/258746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to compute $7^{7^{7^{100}}} \bmod 100$? How to compute $7^{7^{7^{100}}} \bmod 100$? Is $$7^{7^{7^{100}}} \equiv7^{7^{\left(7^{100} \bmod 100\right)}} \bmod 100?$$
Thank you very much.
| First note that $$7^4 \equiv 1 \pmod{100}$$
Hence, we get that $$7^n \equiv \begin{cases}1 \pmod{100} & n \equiv 0 \pmod4\\ 7 \pmod{100} & n \equiv 1 \pmod4\\ 49 \pmod{100} & n \equiv 2 \pmod4\\ 43 \pmod{100} & n \equiv 3 \pmod4 \end{cases}$$
Hence, all we need to figure out is $7^{7^{100}} \pmod 4$. Since $7 \equiv (-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/259870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
compute the following integral $\int^{a}_{-a} \sqrt{a^2-x^2}dx$ I have to compute the following integral:
$\int^{a}_{-a} \sqrt{a^2-x^2}dx$
I did the substitution: $x=a\sin\theta$ so $dx=a\cos\theta d\theta$. The boundaries becomes $\pi/2+2k\pi$ and $-\pi/2-2k\pi$. So:
$\int^{\pi/2+2k\pi}_{-\pi/2-2k\pi} a^2\cos^2\theta ... | Let $I=\int\sqrt{a^2-x^2}dx=\sqrt{a^2-x^2}\int dx-\int\left(\frac{\sqrt{a^2-x^2}}{dx}\int dx\right)dx$
$=x\sqrt{a^2-x^2}-\int\left(\frac{(-2x)x}{2\sqrt{a^2-x^2}}\right)dx$
$=x\sqrt{a^2-x^2}-\int\left(\frac{(a^2-x^2-a^2)}{\sqrt{a^2-x^2}}\right)dx$
$=x\sqrt{a^2-x^2}-\int \sqrt{a^2-x^2}dx +a^2\int\frac{dx}{\sqrt{a^2-x^2}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/260925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
$2^{10 - x} \cdot 2^{10 - x} = 4^{10-x}$ $$2^{10 - x} \cdot 2^{10 - x} = 4^{10-x}$$
Is that correct?
I would've done
$$
2^{10 - x} \cdot 2^{10 - x}\;\; = \;\;
(2)^{10 - x + 10 - x} \; = \; (2)^{2 \cdot (10 - x)} \;=\; 4^{10 - x}\tag{1}
$$
Is that allowed?
If so, can I say that
$$
\frac{4^x}{2^y} = 2^{x - y} \tag{2}
$... | First statement is correct, but $4^x/2^y = 2^{2x-y}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/262176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Help Solving Trigonometry Equation I am having difficulties solving the following equation:
$$4\cos^2x=5-4\sin x$$
Hints on how to solve this equation would be helpful.
|
$\cos^2(x) = 5-4 \sin(x)$
Move everything to the left hand side.
$\cos^2(x)-5+4 \sin(x) = 0$
Write in terms of $sin(x)$ using the identity $\cos^2(x) = 1-\sin^2(x)$:
$4 \sin(x)-4-\sin^2(x) = 0$
Factor constant terms from the left hand side and write the remainder as a square:
$-(\sin(x)-2)^2 = 0$
Multiply both ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
How to evaluate $\int_0^{2\pi} \frac{d\theta}{A+B\cos\theta}$? I'm having a trouble with this integral expression:
$$\int_0^{2\pi} \frac{d\theta}{A+B \cos\theta}$$
I've done this substitution: $t= \tan(\theta/2)$
and get: $\displaystyle \cos\theta= \frac{1-t^2}{1+t^2}$ and $\displaystyle d\theta=\frac{2}{1+t^2}dt$ wher... | To avoid confusion with limits, note that
$$\int_0^{\pi} \dfrac{dx}{a+b \cos(x)} = \int_{\pi}^{2\pi} \dfrac{dx}{a+b \cos(x)}$$
Hence, we have
$$I = \int_0^{2\pi} \dfrac{dx}{a+b \cos(x)} = 2\int_0^{\pi} \dfrac{dx}{a+b \cos(x)}$$
Now use your substitution $t = \tan(x/2)$ and note that $t$ goes from $0$ to $\infty$ as $x$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Can someone please explain to me how I did this summation formula wrong? I was trying to show that
$\sum \limits_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$ but instead I got this $[\frac{n(n+1)}{2}]^2$ which from my understanding I basically proved another summation formula which is $\sum \limits_{k=1}^n k^3$. Obviously I m... | $\sqrt{a^2+b^2}\ne a+b$ in general unless at least one of $a,b$ is $0$
If $s_n=1^2+2^n+\cdots+(n-1)^2+n^2,$
how can you write $s_n=1+2+\cdots+(n-1)+n?$
(1)One way to proof is :
$ (r+1)^3-r^3=3r^2+3r+1$
Put $r=0,1,2,\cdots,n-1,n$ and add to get
$(n+1)^3=3S_n+3(1+2+3+\cdots+n)+n=3S_n+3\frac{n(n+1)}2+n$
So, $S_n=...$
(2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Distinct natural numbers such that $ab=cd=a+b+c+d-3$ Find the distinct natural numbers $a,b,c,d$ who satisfying $ab=cd=a+b+c+d-3$.
| Assume $a$ is the largest number among $a,b,c,d$; then $(a-1)b=a+c+d-3$
$$b=(a+c+d-3)/(a-1)<(a+a+a-3)/(a-1)=3$$
Hence, $b=1$ or $b=2$.
If $b=1$, then $a=a+1+c+d-3$. This implies $c+d=2$. Not the ideal pair.
If $b=2$, then $2a=a+2+c+d-3$. This implies $a=c+d-1$.
$$cd=ab=(c+d-1) \times 2$$
Hence, $$(c-2)(d-2)=2$$
Note th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
for a $3 \times 3$ matrix A ,value of $ A^{50} $ is I f
$$A= \begin{pmatrix}1& 0 & 0 \\
1 & 0 & 1\\
0 & 1 & 0 \end{pmatrix}$$
then $ A^{50} $ is
*
*$$ \begin{pmatrix}1& 0 & 0 \\
50 & 1 & 0\\
50 & 0 & 1 \end{pmatrix}$$
*$$\begin{pmatrix}1& 0 & 0 \\
48 & 1 & 0\\
48 & 0 & 1 \end{pmatrix}$$
*$$\begin{pmatrix}1& 0... | The answer is 3. $$\begin{pmatrix}1& 0 & 0 \\
25 & 1 & 0\\
25 & 0 & 1 \end{pmatrix}$$
Just compute $A^2$ , $A^3$ , $A^4$ and $A^5$ and you will understand the repeated pattern.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/267492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Calculate a double integral I would like to ask a pretty easy question (at least I believe so). I know that:
$$\phi_{11}(k) = \frac{E(k)}{4\pi k^4}(k^2 - k_1^2)$$
$$E(k) = \alpha \epsilon^{\frac{2}{3}}L^{\frac{5}{3}}\frac{k^4}{(1 + k^2)^{\frac{17}{6}}}$$
therefore, substituting the expression of $E(k)$ in $\phi_{1... | The simplest way to approach this is to note that the integrand has radial symmetry: it depends only on $r^2\equiv k_2^2 + k_3^2$. (In other words, use $k^2=k_1^2+k_{\perp}^2$ in the first place, with the appropriate area element.) So the general form here is
$$
\int_{-\infty}^\infty \int_{-\infty}^\infty f(x^2+y^2) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/267784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$3x^2 ≡ 9 \pmod{13}$ What is $3x^2 ≡ 9 \pmod{13}$?
By simplifying the expression as $x^2 ≡ 3 \pmod{13}$ and applying brute force I can show that the answers are 4 and 9, but how to approach this in a more efficient way?
I tried by stating that what the expression above says essentially means $13|(3x²-9)$, which only gi... | We have $13\mid 3(x^2-3)\iff 13\mid(x^2-3)$ as $(3,13)=1$ so, $x^2\equiv3\pmod {13}$.
Now, any number $x$ can be $\equiv 0,\pm1,\pm2,\pm3,\pm4,\pm5,\pm6 \pmod {13}$
So, $x^2\equiv 0,1,4,9,16(\equiv3),25(\equiv 12\equiv-1),36(\equiv10\equiv-3)\pmod {13}$
So, $x\equiv\pm4\pmod {13}$
For a larger prime, we can use Quadrat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Evaluate $\int_0^1{\frac{y}{\sqrt{y(1-y)}}dy}$ I'm a little rusty with my integrals, how may I evaluate the following:
$$
\int_0^1{\frac{y}{\sqrt{y(1-y)}}dy}
$$
I've tried:
$$
\int_0^1{\frac{y}{\sqrt{y(1-y)}}dy} = \int_0^1{\sqrt{\frac{y}{1-y}} dy}
$$
Make the substitution z = 1-y
$$
= \int_0^1{\sqrt{\frac{1-z}{z}} dz}... | Substitute $u=\sqrt{\dfrac{y}{1-y}}.$ Then
$$\dfrac{y}{1-y}=\dfrac{y-1+1}{1-y}=-1+\dfrac{1}{1-y}=u^2, \\
( {0}< {y} <{1} \Leftrightarrow {0}< {u}<{+\infty}),\\
\dfrac{1}{1-y}=1+u^2, \\
1-y=\dfrac{1}{1+u^2}, \\
y=1-\dfrac{1}{1+u^2}, \\
dy=\dfrac{2u}{(1+u^2)^2}\, du,
$$
so
$$\int\limits_0^1{\sqrt{\dfrac{y}{1-y}} dy}=\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
} |
Proving that $\left(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!}\right)$ has a limit $$x_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!}$$
How can we prove that the sequence $(x_n)$ has a limit? I have to use the fact that an increasing sequence has a limit iff it is... | Alternately you can use
$$x_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{n!} \leq x_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+...+\frac{1}{(n-1)\cdot n}$$
and
$$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+...+\frac{1}{(n-1)\cdot n}$$ is telescopic, since
$$\frac{1}{k(k+1)}=\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
How many ways to write one million as a product of three positive integers? In how many ways can the number 1;000;000 (one million) be written as the product
of three positive integers $a, b, c,$ where $a \le b \le c$?
(A) 139
(B) 196
(C) 219
(D) 784
(E) None of the above
This is my working out so far:
$1000000 = 10^{6... | Hint: Solve the problem for $abc = 10^6$. This has $ {8 \choose 2}^2=784$ solutions.
Count the number of solutions where $a=b=c$.
Count the number of solutions where $a=b$ or $b=c$ or $c=a$.
Count the number of solutions where $a, b, c$ are pairwise distinct.
Account for your repeated counting above, to find the cases ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/273137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$(\tan^2(18^\circ))(\tan^2(54^\circ))$ is a rational number Assuming $$\cos(36^\circ)=\frac{1}{4}+\frac{1}{4}\sqrt{5}$$ How to prove that $$\tan^2(18^\circ)\tan^2(54^\circ)$$ is a rational number? Thanks!
| Use the fact that
$$ \tan^2{18^{\circ}} = \frac{1-\cos{36^{\circ}}}{1+\cos{36^{\circ}}} = 1-\frac{2}{5} \sqrt{5} $$
Then use the fact that
$$ \tan^2{54^{\circ}} = \frac{1}{\tan^2{36^{\circ}}} $$
so that
$$ \tan^2{18^{\circ}} \tan^2{54^{\circ}} = \frac{\tan^2{18^{\circ}}}{\tan^2{36^{\circ}}} = \frac{1}{4} (1 -\tan^2{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Calculate $\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$ Please help me calculate this:
$$\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$$
Here I've tried multiplying by $\sqrt[4]{x+9}+2$ and few other method.
Thanks in advance for solution / hints us... | $$\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{\sqrt[4]{x+9}-2}$$
$$=\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{{(x+9)^\frac{1}{4}}-(16)^\frac{1}{4}}.\frac{x-7}{x-7}$$
$$=\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt[3]{x+20}}{x-7}.\underset{x\rightarrow7}{\lim}\frac{x-7}{{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 0
} |
$\displaystyle\sum_{n=2}^{\infty}\frac{2}{(n^3-n)3^n} = -\frac{1}{2}+\frac{4}{3}\sum_{n=1}^{\infty}\frac{1}{n\cdot 3^n}$ Please help me, to prove that
$$
\sum_{n=2}^{\infty}\frac{2}{(n^3-n)3^n} =
-\frac{1}{2}+\frac{4}{3}\sum_{n=1}^{\infty}\frac{1}{n\cdot 3^n}.
$$
| Maybe we wanna use the fact that $\displaystyle \sum_{n=1}^{\infty}\frac{1}{ns^n}=\ln\frac{s}{s-1}, \space s>1$. Then
$$\sum_{n=2}^{\infty} \frac {2}{(n^3-n)3^n}=\frac{1}{3}\sum_{n=2}^\infty \frac {1}{(n-1)3^{n-1}} - 2\sum_{n=2}^\infty \frac {1}{n3^n} + 3\sum_{n=2}^\infty \frac {1}{(n+1)3^{n+1}}=\frac{4}{3} \log\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Proof of tangent half identity
Prove the following: $$\tan \left(\frac{x}{2}\right) = \frac{1 + \sin (x) - \cos (x)}{1 + \sin (x) + \cos (x)}$$
I was unable to find any proofs of the above formula online. Thanks!
| You could always use Euler's formula, setting $$ z = e^{i x} $$ which gives for the left
$$ \tan\left(\frac{x}{2}\right) = \frac{1}{i} \frac{z^{1/2}-z^{-1/2}}{z^{1/2}+z^{-1/2}}
= \frac{1}{i} \frac{z-1}{z+1}$$
and for the right
$$ \frac{1+\sin(x)-\cos(x)}{1+\sin(x)+\cos(x)} =
\frac{1+ 1/(2i) z - 1/(2i) 1/z - (1/2) z - (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/277106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Finding $\frac{1}{S_1}+\frac{1}{S_2}+\frac{1}{S_3}+\cdots+\frac{1}{S_{2013}}$ Assume $S_1=1 ,S_2=1+2,S=1+2+3+,\ldots,S_n=1+2+3+\cdots+n$
How to find :
$$\frac{1}{S_1}+\frac{1}{S_2}+\frac{1}{S_3}+\cdots+\frac{1}{S_{2013}}$$
| Hint 1:
$$ \sum_{k=1}^n k = \frac{n (n+1)}{2} $$
Hint 2:
$$ \frac{1}{n (n+1)} = \frac{1}{n} - \frac{1}{n+1} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/281318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
The number of ways of completing this partial Latin square If we want to fill the empty squares by the numbers $1$, $2$, $3$, $4$, $5$, $6$ so that all the numbers appear in each row and column, how can we find the number of ways to do that?
$$\begin{array}{|c|c|c|c|c|c|}
\hline
\;1\strut\;& \;2\; & \;3\; & \;4\; & \;5... | The second row must contain a $3$ and a $4$, neither of which can be in either the third or the fourth column; thus, they must be in the second and fifth columns, in either order, and the missing $1$ and $6$ can then be filled in in either of two ways. That is, there are $2^2=4$ ways to fill in the second row acceptabl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find a transformation in specified basis My task is to find a matrix of linear transformation $\varphi$ in basis $A,B$
$\varphi:\mathbb{R}^{2}\to\mathbb{R}^{4}
\varphi((x_{1},x_{2}))=(3x_{1}+x_{2},x_{1}+5x_{2},-x_{1}+4x_{2},2x_{1}+x_{2})$
$\mathcal{A}=\{(3,1),(4,2)\}
\mathcal{B}=\{(1,0,1,0),(0,1,1,1),(0,1,2,3),(0,0... | Assuming that vectors in $\mathbb{R}^2$ and $\mathbb{R}^4$ are represented by column vectors, you should find $M_B^{st}(id)^{-1}M_{st}^{st}(\varphi)M_A^{st}(id)$ instead. If you adopt a row vector convention, just transpose the resulting matrix.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/282390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Let $a,b$ and $c$ be real numbers.evaluate the following determinant: |$b^2c^2 ,bc, b+c;c^2a^2,ca,c+a;a^2b^2,ab,a+b$| Let $a,b$ and $c$ be real numbers. Evaluate the following determinant:
$$\begin{vmatrix}b^2c^2 &bc& b+c\cr c^2a^2&ca&c+a\cr a^2b^2&ab&a+b\cr\end{vmatrix}$$
after long calculation I get that the answer w... | If $b=0,$
$$\begin{vmatrix}b^2c^2 &bc& b+c\cr c^2a^2&ca&c+a\cr a^2b^2&ab&a+b\cr\end{vmatrix} =\begin{vmatrix}0 &0&c\cr c^2a^2&ca&c+a\cr 0&0&a\cr\end{vmatrix}$$
Now, if $a=0,$ $$\text{the determinant becomes }\begin{vmatrix}0 &0&c\cr 0&0&c\cr 0&0&0\cr\end{vmatrix}=0$$
else for $ca\ne 0$ $$\text{the determinant becomes ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Problem related to a square matrix Let $A$ be an $n\times n$ matrix with real entries such that $A^{2}+I=\mathbf{0}$. Then:
(A) $n$ is an odd integer.
(B) $n$ is an even integer.
(C) $n$ has to be $2$
(D) $n$ could be any positive integer.
I was thinking about the problem.I noticed for a $2\times 2$ matrix $A$ of ... | Note:
$n=1\,$ is ruled out, since e.g., $A$ consists of the single scalar entry $1$: $A = [1],\; I = I_1,\;, A^2 + I = 2.\;$
Indeed there is no real scalar $\,k\,\neq 0\,$ (in the case $n = 1, A = k\,$) such that $\,k^2 = -1.\,$
So option (A) is ruled out, since $\,n = 1\,$ is odd, and option (D) is ruled out, since... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/283444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Consider a function $f(x) = x^4+x^3+x^2+x+1$, where x is an integer, $x\gt 1$. What will be the remainder when $f(x^5)$ is divided by $f(x)$? Consider a function $f(x) = x^4+x^3+x^2+x+1$, where x is an integer, $x\gt 1$. What will be the remainder when $f(x^5)$ is divided by $f(x)$ ?
$f(x)=x^4+x^3+x^2+x+1$
$f(x^5)=x^{2... | Let $x^{20}+x^{15}+x^{10}+x^5+1=(x^4+x^3+x^2+x+1)Q(x)+R(x)$.
$x^4+x^3+x^2+x+1=0$ has $4$ complex roots $a_1,a_2,a_3,a_4$. And these are also roots of $x^5=1$, so when $x=a1,a2,a3,a4$, the above equation becomes
$5=R(a_i)$ ($i=1,2,3,4$)
This is true when $R(x)=5$ for every $x$, and it is easy to show
that a polynomial o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$ How can I find the formula for the following equation?
$$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$$
More importantly, how would you approach finding the formula? I have fo... | Hint: Use the fact that $\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$ and find $S_n=\sum_1^n\left(\frac{1}{k}-\frac{1}{k+1}\right)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/286024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 1
} |
Proving that $ f(1)=\frac{1-\sqrt{5}}{2}$ for this function
Let $f:(0,+\infty)\mapsto R$ be a strictly increasing function such that $\forall x\ge0,$ $$f(x)+\frac{1}{x}\ge0, \qquad f(x)f\left(f(x)+\frac{1}{x}\right)=1.$$
Show that $$f(1)=\frac{1-\sqrt{5}}{2}.$$
Please give an example that satisfies these condit... | $f(x)f(f(x)+\frac{1}{x})=1$
$x=1 $ gives $ f(1)f(f(1)+1)=1$
$x=f(1)+1\ge0$ gives $ f(f(1)+1)f(f(f(1)+1)+\frac{1}{f(1)+1})=1$
By replacing $f(f(1)+1)$ by $\frac{1}{f(1)}$ you get $\frac{1}{f(1)}f(\frac{1}{f(1)}+\frac{1}{f(1)+1})=1$
Multiply both sides by $f(1)$ to get $f(\frac{1}{f(1)}+\frac{1}{f(1)+1})=f(1)$
Then you k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/286932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Equations over permutations Let $\sigma=\begin{pmatrix} 1 & 2 & 3 & 4 \\ 3 & 1 & 4 & 2 \end{pmatrix} \in S_4$ and $\theta=\begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{pmatrix} \in S_4$.
Solve the following equations $(x \in S_4)$:
a) $x \sigma = \sigma x$;
b) $x^2 = \sigma$;
c) $x^2 = \theta$.
I'm writing here... | a) Let $x=\begin{pmatrix} 1 & 2 & 3 & 4 \\ a & b & c & d \end{pmatrix} \in S_4$. Then
$x\sigma=\begin{pmatrix} 1 & 2 & 3 & 4 \\ a & b & c & d \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 & 3 & 4 \\ 3 & 1 & 4 & 2 \end{pmatrix}=\begin{pmatrix} 1 & 2 & 3 & 4 \\ c & a & d & b \end{pmatrix}$.
We conclude that $x(\sigma(1))=c$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/287339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find five positive integers whose reciprocals sum to $1$ Find a positive integer solution $(x,y,z,a,b)$ for which
$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$
Is your answer the only solution? If so, show why.
I was surprised that a teacher would assign this kind of problem to a 5th gr... | Note that $\frac{4!}{4!}=1$. Now write $4!=24$ as $1+2+3+6+12$ (basically the divisors of $24$) Then we have $$1=\frac{4!}{4!}=\frac{1+2+3+6+12}{24}=\frac{1}{24}+\frac{1}{12}+\frac{1}{8}+\frac{1}{4}+\frac{1}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/290435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "373",
"answer_count": 16,
"answer_id": 14
} |
Compute the limit $\displaystyle \lim_{x\rightarrow 0}\frac{n!.x^n-\sin (x).\sin (2x).\sin (3x).......\sin (nx)}{x^{n+2}}\;\;,$ How can i calculate the Given limit
$\displaystyle \lim_{x\rightarrow 0}\frac{n!x^n-\sin (x)\sin (2x)\sin (3x)\dots\sin (nx)}{x^{n+2}}\;\;,$ where $n\in\mathbb{N}$
| Since $\sin x = x - x^3/6 +O(x^5)$ as $x\to 0$, we get
$$\begin{array}
. & &\frac{n!x^n-\sin (x)\sin (2x)\sin (3x)\cdots\sin (nx)}{x^{n+2}}
\\&=&\frac{n!x^n - (x-x^3/6+O(x^5))\cdots(nx-(nx)^3/6+O(x^5))}{x^{n+2}}
\\&=& \frac{\frac{1}{6}x^{n+2}n! (1^2+2^2+\cdots+n^2)+O(x^{n+4})}{x^{n+2}}
\end{array}$$
as $x\to0$.
So des... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/291087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Computing the value of $\operatorname{Li}_{3}\left(\frac{1}{2} \right) $
How to prove the following identity
$$ \operatorname{Li}_{3}\left(\frac{1}{2} \right) = \sum_{n=1}^{\infty}\frac{1}{2^n n^3}= \frac{1}{24} \left( 21\zeta(3)+4\ln^3 (2)-2\pi^2 \ln2\right)\,?$$
Where $\operatorname{Li}_3 (x)$ is the trilogarithm... | We can find the value similarly as seen here for $\operatorname{Li}_2\left(\frac12\right)=\frac{\pi^2}{12}-\frac{\ln^2 2}{2}$.
From the question we have:
$$\operatorname{Li}_3\left(\frac12\right)=\sum_{n=1}^\infty \frac{1}{n^3}\frac{1}{2^n}=\frac12 \int_0^1 \frac{\ln^2 x}{2-x}dx\overset{\large \frac{x}{2-x}=t}=\frac12\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/293724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
} |
If $A\equiv 1\pmod{3}$, then $4p=A^2+27B^2$ uniquely determines $A$. If $p\equiv 1\pmod{3}$, it's well know that $p$ can be expressed as
$$
p=\frac{1}{4}(A^2+27B^2).
$$
In this letter by Von Neumann, he mentions that Kummer determined that $A$ is in fact uniquely determined by the additional condition $A\equiv 1\pmod{3... | It looks like you're doing calculation in the number field $\mathbb{Q}(\sqrt{-3})$. In particular, the typical algebraic integer is of the form
$$ u + v \frac{1 + \sqrt{-3}}{2} $$
or put differently, as
$$ \frac{A + B \sqrt{-3}}{2} $$
where $A$ and $B$ have the same parity. The norm of this element is
$$ N = \frac{A + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/295854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Gaussian-Jordan Elimination question? I have the linear system
$$
\begin{align*}
2x-y-z+v&=0 \\
x-2y-z+5u-v&=1 \\
2x-z+v&=1
\end{align*}$$
Very well. I form the matrix
$$
\left[
\begin{array}{@{}ccccc|c@{}}
2&-1&-1 & 0 & 1 &0 \\
1&-2&-1 & 5 & -1 &1 \\
2&0&-1 & 0&1&1 \\
\end{array}
\right]
$$
So I thought about exch... | Assuming that you've done everything right so far
$$
\left[
\begin{array}{@{}ccccc|c@{}}
-2&2&2 & -10 & -4 &-2 \\
0&-3&1 &-10 & -5 &-2 \\
0&0&0& 0&1&1 \\
\end{array}
\right]
$$
Now, let the first row be L1, the second L2, the third L3.
L1$\times$(-0.5) and then, L1-2L2.
You got
$$
\left[
\begin{array}{@{}ccccc|c@{}}
1&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/298128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding Binomial expansion of a radical I am having trouble finding the correct binomial expansion for $\dfrac{1}{\sqrt{1-4x}}$:
Simplifying the radical I get: $(1-4x)^{-\frac{1}{2}}$
Now I want to find ${n\choose k} = {\frac{-1}{2}\choose k}$
\begin{align}
{\frac{-1}{2}\choose k} &= \dfrac{\frac{-1}{2}(\frac{-1}{2}-1)... | Also, generalizing what Andre Nicolas wrote,
\begin{align}
{\frac{-1}{2}\choose k} &= \dfrac{\frac{-1}{2}(\frac{-1}{2}-1)(\frac{-1}{2}-2)\ldots(\frac{-1}{2}-k+1)}{k!} \\
&= (-1)^k\dfrac{\frac{1}{2}(\frac{1}{2}+1)(\frac{1}{2}+2)\ldots(\frac{1}{2}+k-1)}{k!} \\
&=(-1)^k \frac{1\ 3\ 5 ... (2k-1)}{2^k k!} \\
&=(-1)^k \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/299421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$ Inspired by Ramanujan's problem and solution of $\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \ldots}}}$, I decided to attempt evaluating the infinite radical
$$
\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}
$$
Taking a cue from Ramanujan's solution method, I defi... | Since you have recurrence relation $(f(x))^2=2^x+f(x+1)$ you could find an approximate solution by approximating $f(x+1)\approx f(x)$ and then you get quadratic equation for the function $f$ and in doing so you can find an approximate value $f(x)$ for every value of $x$. It is clear from the recurrence relation that th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 6,
"answer_id": 3
} |
Find all pairs of positive whole numbers Find all pairs of positive whole numbers x and y which are a solution for $ \dfrac{2}{x} + \dfrac {3}{y} = 1 $.
I don't really understand how to tackle this question. I rewrote $ \dfrac{2}{x} + \dfrac {3}{y} = 1 $ as $2y+ 3x =xy$ but that's it..
| If we multiply both sides the original equation by $xy$, we ge $2y+3x=xy$. We can rewrite this as $2y+3x-xy=0$. We now perform a little trick.
Note that what we have is very much like $(3-y)(x-2)=3x+2y-xy-6$. If we subtract $6$ from both sides, we get $2y+3x-xy-6=-6$ and $(3-y)(x-2)=-6$. Multiplying through by $-1$ giv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
How to solve a system of 3 equations with Cramer's Rule? I am given the following system of 3 simultaneous equations:
$$
\begin{align*}
4a+c &= 4\\
19a + b - 3c &= 3\\
7a + b &= 1\end{align*}
$$
How do I solve using Cramers' rule?
For one, I do know that by putting as a matrix the LHS
$$\begin{pmatrix}
4&0&1\\19&1&-3\\... | $$\begin{pmatrix}
4&0&1\\19&1&-3\\7&1&0
\end{pmatrix}=\frac{1}{a}\begin{pmatrix}
4a&0&1\\19a&1&-3\\7a&1&0
\end{pmatrix}$$
$$\Rightarrow \frac{1}{a}\begin{pmatrix}
(4a+0.b+1.c)&0&1\\(19a+b-3c)&1&-3\\(7a+b+0.c)&1&0
\end{pmatrix}$$
$$\Rightarrow \frac{1}{a}\begin{pmatrix}
4&0&1\\3&1&-3\\1&1&0
\end{pmatrix}$$
$$\Rightarrow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/304615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$ Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac 32x_{1}}\left(\... | The expression equals to $2n$ when all $x_i=\frac13$.
The expression cannot be less than $2n$ because of AM-GM inequality due to the following:
$$\frac{\ln\left(\frac12-\frac1{36x^2}\right)}{\ln\left(\frac32x\right)}=2\frac{\ln\left(\frac94x^2-\left(\frac32x-\frac1{6x}\right)^2\right)}{\ln\left(\frac94x^2\right)}\ge2$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$ Prove that
$$\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$$
EDIT: inspired by Michael Hardy's suggestion I got that
$$\arcsin \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{(n+1)(n+2)}}=\arc... | Since $\arcsin x\ge \arctan x$ for $x \in [0,1]$, thus we shall have
$$\arcsin(\frac{\sqrt{n+1}-\sqrt{n}}{n+1})\ge \arctan(\frac{\sqrt{n+1}-\sqrt{n}}{n+1})\ge \arctan\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n+1}\sqrt{n}+1} $$ $$= \arctan{\sqrt{n+1}}-\arctan{\sqrt{n}}.$$
(The last equality uses $\arctan(x) - \arctan(y) = \arcta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/305690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
} |
Integral of type $\displaystyle \int\frac{1}{\sqrt[4]{x^4+1}}dx$ How can I solve integral of types
(1) $\displaystyle \int\dfrac{1}{\sqrt[4]{x^4+1}}dx$
(2) $\displaystyle \int\dfrac{1}{\sqrt[4]{x^4-1}}dx$
| Looks like the variable substitution in Lab Bhattacharjee's answer can be generalized to handle indefinite integrals of the form:
$$ \int \frac{dx}{\sqrt[n]{x^{n}+1}}$$
Let $y = \frac{x}{\sqrt[n]{x^n+1}}$, we have:
$$n y^{n-1} dy = \frac{n x^{n-1} dx}{(1+x^n)^2} \implies \frac{dy}{y} = \frac{dx}{x(1+x^n)} = ( 1 - y^n)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$ The following question comes from Some integral with sine post
$$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$
but now I'd be curious to know how to deal with it by methods of complex analysis.
Some suggestions, hints? ... | I have a generalized elementary method for this problem，If f (x) is an even function, and the period is $\pi$,we have:
$$\int_{0}^\infty f(x)\frac{\sin^nx}{x^n}dx=\int_{0}^\frac{\pi}{2}f(x)g_n(x)\sin^nxdx \qquad (1)$$
Where the $g_n(x)$ in (1) is as follows
$$g_n(x)=\begin{cases}\frac{(-1)^{n-1}}{(n-1)!}\frac{d^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 6,
"answer_id": 1
} |
Need help using De Moivre's theorem to write $\cos 4\theta$ & $\sin 4\theta$ as terms of $\sin\theta$ and $\cos\theta$ I need help with the following question:
"Use De Moivre's theorem to write $\cos 4\theta$ & $\sin 4\theta$ as terms of $\sin\theta$ and $\cos\theta$"
You could write the problem as:
$(\cos\theta+i\sin... | Use the binomial theorem for $(x+i y)^4$:
$$\begin{align}(x+i y)^4 &= x^4 + 4 i x^3 y +6 i^2 x^2 y^2+4 i^3 x y^3 + i^4 y^4\\&= x^4-6 x^2 y^2 + y^4 + i (4 x^3 y - 4 x y^3) \end{align}$$
using $i^2=-1$, etc. Now let $x=\cos{\theta}$, $y=\sin{\theta}$:
$$e^{i 4 \theta} = \cos{4 \theta} + i \sin{4 \theta} = (\cos{\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Does $\frac{3}{1\cdot 2} - \frac{5}{2\cdot 3} + \frac{7}{3\cdot 4} - ...$ Converges? $$\frac{3}{1\cdot 2} - \frac{5}{2\cdot 3} + \frac{7}{3\cdot 4} - ...$$
Do you have an idea about this serie? If it converges what is the sum?
| This is
$$\sum_{k=1}^{\infty} (-1)^{k+1} \frac{2 k+1}{k (k+1)}$$
For large $k$, the summand behaves as $2(-1)^k/k$. By comparison to
$$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} = \log{2}$$
the sum converges.
Note that the sum may be expressed as
$$\sum_{k=1}^{\infty} (-1)^{k+1} \left ( \frac{1}{k} + \frac{1}{k+1} \righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Finding the derivative of $x$ to the power something that is a function of $x$ if $y = x^{(x+1)^\frac12}$
then how can I get the first derivative of $y$?
| Another way to solve it is to take $\ln$ of both sides, and apply implicit differentiation:
$$y=x^{(x+1)^{\frac {1}{2}}}$$
$$\ln y=\ln x^{(x+1)^{\frac {1}{2}}}$$
Rewriting using $\log$ properties:
$$\ln y=(x+1)^{\frac {1}{2}} \cdot \ln x$$
Now take the derivative implicitly:
$$\frac{1}{y}\cdot y' = \frac{1}{2}(x+1)^{-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How does this sum go to $0$? http://www.math.chalmers.se/Math/Grundutb/CTH/tma401/0304/handinsolutions.pdf
In problem (2), at the very end it says
$$\left(\sum_{k = n+1}^{\infty} \frac{1}{k^2}\right)^{1/2} \to 0$$
I don't see how that is accomplished. I understand the sequence might, but how does the sum $$\left ( \fr... | Since the sum $\sum_{k=1}^\infty \frac{1}{k^2}$ is convergent, you can make the "tail" of the sum ($\sum_{k=n+1}^\infty \frac{1}{k^2}$) as small as you want.
Explicitly, since $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$, for any $\epsilon>0$, you can find an $n$ such that $\sum_{k=n+1}^\infty \frac{1}{k^2}=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 4
} |
Maclaurin series for $\frac{x}{e^x-1}$ Maclaurin series for
$$\frac{x}{e^x-1}$$
The answer is
$$1-\frac x2 + \frac {x^2}{12} - \frac {x^4}{720} + \cdots$$
How can i get that answer?
| One way is to write $e^x-1 $ as $1 + x + x^2/2 + ... - 1$ and then factor out $x$ and cancel up the top and expand it as geometric series and collect the coefficients of like powers.
$\displaystyle
\begin{align*}
e^x - 1 &= x + \frac{x^2}{2!} + \frac{x^3}{3!} + o(x^4)\\
\frac{x}{e^x - 1} &= \frac{1}{1 + \left( \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 0
} |
Determining a point's coordinates on a circle So I have a circle (I know its center's coordinates and radius) and a point on the circle (I know its coordinates) and I have to determine the coordinates of another point on the circle which is exactly at the distance L from the first point.
| If the equation of the circle be $(x-h)^2+(y-k)^2=r^2$ any point on the circle can be parametrized as $(r\cos\theta+h,r\sin\theta+k)$ and if the known point on the circle be $S(p,q)$
So, we need to know the two values of $\theta$ from
$(r\cos\theta+h-p)^2+(r\sin\theta+k-q)=L^2$
$\implies 2r(h-p)\cos\theta+2r(k-q)\sin\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/312704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Rolling 2 fair, 6 sided dice find P(sum=12, 2times in 36 rolls) I have a dice problem.
If we roll 2 fair dice, and the sum is 12 then our test is a pass, otherwise its a fail.
What is the probability that the number of passes in 36 tests is greater then 1.
S={1,2,3,4,5,6}
(S,S) = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5... | Note that there is only way way to achieve a $12$ when rolling two dice (whose outcomes we will denote by the random variables $X_{1}$ and $X_{2}$) and taking their sum, which is when both dice show $6$, which are both independent events with probability of $\frac{1}{6}$, therefore:
$$P(X_{1}+X_{2}=12)=P(X_{1}=6 \cap ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simple AM-GM inequality Let $a,b,c$ positive real numbers such that $a+b+c=3$, using only AM-GM inequalities show that
$$
a+b+c \geq ab+bc+ca
$$
I was able to prove that
$$
\begin{align}
a^2+b^2+c^2 &=\frac{a^2+b^2}{2}+\frac{b^2+c^2}{2}+\frac{a^2+c^2}{2} \geq \\
&\ge \frac{2\sqrt{a^2b^2}}{2}+\frac{2\sqrt{b^2c^2}}{2}+\... | $9 = (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab+bc+ac) \geq 3(ab+bc + ac)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/315699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 3
} |
Two finite fields are isomorphic. Let $F = \Bbb{Z}_2$. Given the irreducible polynomials $f(x)= x^3 + x + 1$, and $g(y) = y^3 + y^2 + 1$, form the fields $K = F[x]/(f(x))$ and $E = F[y] / (g(y))$. These are fields of order 8 (given), so they must be isomorphic. Is the map $[x] \mapsto [y + 1]$ an isomorphism? It's c... | An element of $K$ has a unique representation as $p + (f)$, where $\deg p \leq 2$ and an element of $E$ has a unique representation as $q + (g)$, where $\deg q\leq 2$. Take $A = ax^2 + bx + c + (f)$, $B = a'x^2 + b'x + c + (f)$.
\begin{align*}
\phi(A + B) &= \phi(ax^2 + bx + c + (f) + a'x^2 + b'x + c + (f))\\
&= \phi((... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Proving the inequality $\arctan\frac{\pi}{2}\ge1$ Do you see any nice way to prove that
$$\arctan\frac{\pi}{2}\ge1 ?$$
Thanks!
Sis.
| I'm not sure if this would be considered nice, but anyway:
It suffices to show that $\tan{1} \leq \frac{\pi}{2}$, or equivalently $\sin{1} \leq \frac{\pi}{2}\cos{1}$.
$$\sin{x}=(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!})+\sum_{n=2}^{\infty}{\left(\frac{x^{4n+1}}{(4n+1)!}-\frac{x^{4n+3}}{(4n+3)!}\right)}$$
$$\cos{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 3
} |
Problem involving permutation matrices from Michael Artin's book. Let $p$ be the permutation $(3 4 2 1)$ of the four indices. The permutation matrix associated with it is
$$ P =
\begin{bmatrix}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0
\end{bmatrix}
$$
This is the matrix that permutes the comp... | The problem is that you are multiplying cycles from left to right, but matrices multiply from right to left.
Let's take this out of cycle notation for a second. $$(3421)=\left(\begin{array}{cccc}1&2&3&4\\3&1&4&2\end{array}\right)$$
You want to say that $$(3421)=(12)(14)(13).$$ If you are multiplying right to left, it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
$2005|(a^3+b^3) , 2005|(a^4+b^4 ) \implies2005|a^5+b^5$ How can I show that if $$2005|a^3+b^3 , 2005|a^4+b^4$$ then $$2005|a^5+b^5$$
I'm trying to solve them from $a^{2k+1} + b^{2k+1}=...$ but I'm not getting anywhere.
Can you please point in me the correct direction?
Thanks in advance
| Actually you don't need the $a^4 + b^4$.
$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$. It turns out that $a^2 - ab + b^2 \equiv 0$ has no solutions except $(0,0)$ mod either $5$ or $401$, so the only way to have $a^3 + b^3 \equiv 0 \mod 2005$ is
$a + b \equiv 0$, and then you also have $a^k + b^k \equiv 0 \mod 2005$ for every ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/319248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
} |
An Inequality question I have the following question. I have to find a $\delta>0$ such that for all complex numbers $x,y$ the following holds true - \begin{equation}
\frac{1}{2\pi}\int_0^{2\pi}|x+e^{it}y|\,dt \ge (|x|^2+\delta|y|^2)^{1/2}.
\end{equation}
I have proceeded in the following way. Clearly, if $x=0$, then t... | I think the inequality is not correct if we are looking at 2-uniform PL convex (see http://poincare.matf.bg.ac.rs/~pavlovic/BLAsco.PDF page 750) .
The case if a power of 2 should be on the integrad and no square root on the RHS.
Note if $y=0$ the equality holds for all $\delta \in \mathbb{R}$. Hence let $y \neq 0$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/320240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
prove $ \frac{1}{13}<\frac{1}{2}\cdot \frac{3}{4}\cdot \frac{5}{6}\cdot \cdots \cdot \frac{99}{100}<\frac{1}{12} $ Without the aid of a computer，how to prove
$$ \frac{1}{13}<\frac{1}{2}\cdot \frac{3}{4}\cdot \frac{5}{6}\cdot \cdots \cdot \frac{99}{100}<\frac{1}{12} $$
| Another approach, perhaps simpler:
Let $X = \frac{1}{2}\cdot \frac{3}{4}\cdot \frac{5}{6}\cdot \cdots \cdot \frac{99}{100}$
We need to show $\dfrac{1}{13} < X < \dfrac{1}{12}$, which is equivalent to showing $144 < \dfrac{1}{X^2} < 169$, and we shall proceed to do this.
For the lower bound, note that we need to s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/322224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
} |
Prove that only parallelogram satisfies these conditions
Sum of distances between middle points of two opposite sides of a quadrilateral is equal to its semiperimeter. Prove that the quadrilateral has to be parallelogram.
I have no idea where to start. I tried with using middle line of triangle, messed around with ar... | Hint: Use vector notation. Write out both sides in terms of the position vectors of $A, B, C, D$.
Hint: Apply the triangle inequality to show that equality must occur.
Hint: Equality only holds when the 3 vertices of a triangle are in a straight line, giving us the parallel condition that we desire.
(Abusing vector n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there an easy way to calculate $\lim\limits_{k \to \infty} \frac{(k+1)^5(2^k+3^k)}{k^5(2^{k+1} + 3^{k+1})}$? Is there an easy way to calculate $$\lim_{k \to \infty} \frac{(k+1)^5(2^k+3^k)}{k^5(2^{k+1} + 3^{k+1})}$$
Without using L'Hôpital's rule 5000 times?
Thanks!
| It is a product of the following two expressions
$\frac{(k+1)^5}{k^5}=\left(1+\frac1k\right)^5$
$\frac{2^k+3^k}{2^{k+1}+3^{k+1}}= \frac{3^k(1+(2/3)^k)}{3^{k+1}(1+(2/3)^{k+1})}= \frac13\cdot$ $\frac{1+(2/3)^k}{1+(2/3)^{k+1}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/327325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Need help finding unknowns in simplex tableau. I need help with this homework problem.
The objective is to maximize $2x_1 - 4x_2$, and the slack variables are $x_3$ and $x_4$. The constraints are $\le$ type.
Tableau
$\begin{matrix}z & x_1 & x_2 & x_3 & x_4 & \text{RHS}\\
1 & b & 1 & f & g & 8\\
0 & c & 0 & 1 & 1\over... | Additional notations
*
*$A \in M_{m\times n}(\Bbb R)$ ($m\le n$) has rank $n$ and basis matrix $B$.
*$x_B$ denotes the basic solution.
*$c_B$ denotes the reduced objective function so that $c^T x=c_B^T x_B$. (So the order/arrangement of basic variables is very important.)
Unknown entries in the tableau
*
*Fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/328077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How do I transform the left side into the right side of this equation? How does one transform the left side into the right side?
$$
(a^2+b^2)(c^2+d^2) = (ac-bd)^2 + (ad+bc)^2
$$
| $$(a^2+b^2)(c^2+d^2)$$
$$=a^2.c^2+a^2.d^2+b^2.c^2+b^2.d^2$$
$$=a^2.c^2+b^2.d^2+a^2.d^2+b^2.c^2$$
$$=a^2.c^2-2a.b.c.d+b^2.d^2+a^2.d^2+2a.b.c.d+b^2.c^2$$
$$=(ac)^2 - 2.(ac).(bd)+(bd)^2 + (ad)^2 + 2(ad)(bc) + (bc)^2$$
$$=(ac - bd)^2+(ad+bc)^2$$
$$=R.H.S$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/330863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 1
} |
Half-symmetric, homogeneous inequality Let $x,y,z$ be three positive numbers. Can anybode prove the follwing inequality :
$(x^2y^2+z^4)^3 \leq (x^3+y^3+z^3)^4$ (or find a counterexample, or find a reference ...)
| We need to show $(x^2y^2 + z^4)^3 \le (x^3+y^3+z^3)^4.$
By AM-GM, we have $(x^3+y^3+z^3)^4 \ge \left(2(xy)^{\frac{3}{2}} + z^3\right)^4$
Let $a = \sqrt{xy} > 0$. Then it is sufficient to show that
$(2a^3 + z^3)^4 \ge (a^4 + z^4)^3$
Let $t = \frac{a}{z} > 0$, then we need to show
$f(t) = (2t^3 + 1)^4 - (t^4 + 1)^3 \g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Simplify $ \frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}} $ Please help me find the sum
$$
\frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}}
$$
| Adding the terms together, you should get:
$$
\sum_{\text{all terms}} = \frac{(32 x^{31})}{(x^{32}-y^{32})}
$$
This result is obtained by using the LCD, (Least Common Denominator).
LCD:
$$
(x-y) (x+y) (x^2+y^2) (x^4+y^4) (x^8+y^8) (x^{16}+y^{16}) =\\ (x^2-y^2)(x^2+y^2)(x^4+y^4) (x^8+y^8) (x^{16}+y^{16})=\\
(x^4-y^4)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/332191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Showing divergence of the series. I am having hard time trying to show that the following series is divergent. Can someone help me please?
$$\sum \frac{(-1)^n}{\log n} b_n $$
where $b_n=\frac{1}{\log n} $ if n is even
and $b_n=\frac{1}{2^n}$ if n is odd.
This would be a great help!
| Your series is:
$$
\frac{1}{\log{2}}\frac{1}{\log{2}} - \frac{1}{\log{3}}\frac{1}{2^3} + \frac{1}{\log{4}}\frac{1}{\log{4}}-\frac{1}{\log{5}}\frac{1}{2^5}+-\ldots
$$
Which is:
$$
\sum_{2}^{\infty}\left(\frac{1}{\log{2n}}\right)^2 - \sum_{2}^{\infty} \frac{1}{\log{(2n+1)}2^{2n+1}}
$$
$$
\sum_{2}^{\infty} \frac{1}{\lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/332253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Logarithm simplification Simplify: $\log_4(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}})$
Can we use the formula to solve this: $\sqrt{a+\sqrt{b}}= \sqrt{\frac{{a+\sqrt{a^2-b}}}{2}}$
Therefore first term will become: $\sqrt{\frac{3}{2}}$ + $\sqrt{\frac{1}{2}}$
$\log_4$ can be written as $\frac{1}{2}\log_2$
Please guide further.... | Hint:
$(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}})=\sqrt k$
Your $k=6$, Now its just $\frac{1}{2} \log_46$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/332636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Derivative of $\ln\left(\frac{2x}{1+x}\right)$ I know that the derivative of
$$f(x)=\ln(x) \ ,\ x>0$$
is just simply
$$f'(x)=\frac{dx}{x}$$
But how do you find the derivative for the function:
$$g(x)=\ln\left(\frac{2x}{1+x}\right)\ , \ x>0$$
| The trick hinted at by Peter Tamaroff in another answer is a clever one, but in general you use the chain rule to find the derivative of a composition:
$$(f(g(x)))' = f'(g(x))\cdot g'(x).$$
In your case, $f(x)=\ln x$ and $g(x)=2x/(1+x)$. Let us first find the derivatives of $f$ and $g$:
\begin{align*}
f'(x) &= (\ln x)'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Calculating $\lim_{x \rightarrow 1}(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}})$ How to calculate following limit?
$$\lim_{x \rightarrow 1}\left(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}}\right)$$
| Let $x=1+y$, then
$$
\begin{align}
\frac{23}{1-x^{23}}-\frac{31}{1-x^{31}}
&=\frac{23}{1-(1+23y+\frac{23\cdot22}{2\cdot1}y^2+O(y^3))}\\
&-\frac{31}{1-(1+31y+\frac{31\cdot30}{2\cdot1}y^2+O(y^3))}\\
&=-\frac1y(1-\tfrac{22}{2\cdot1}y+O(y^2))\\
&\hphantom{=}+\frac1y(1-\tfrac{30}{2\cdot1}y+O(y^2))\\
&=(\tfrac{22}{2\cdot1}-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
Show that $x_n \leq \frac{1}{\sqrt{3n+1}}$ Let $$x_n=\frac{1}{2} \frac{3}{4}\frac{5}{6}\cdots\frac{2n-1}{2n}$$
Then show that $$x_n \leq \frac{1}{\sqrt{3n+1}}$$ for all $n=1,2,3,\dots$
I try induction but unable to solve this equality.
| If (induction)
$$
x_n \leq \frac{1}{\sqrt{3n +1}}
$$
Then
$$
x_{n+1} = \frac{1}{2} \frac{3}{4}\frac{5}{6}\cdots\frac{2n-1}{2n}\frac{2n+1}{2n+2} \leq \frac{1}{\sqrt{3n+1}}\frac{2n+1}{2n+2}
$$
You want now to prove that
$$
\frac{1}{\sqrt{3n+1}}\frac{2n+1}{2n+2} \leq \frac{1}{\sqrt{3n+4}}.
$$
That is:
$$
\sqrt{3n+4}(2n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find a closed form of the series $\sum_{n=0}^{\infty} n^2x^n$ The question I've been given is this:
Using both sides of this equation:
$$\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$$
Find an expression for $$\sum_{n=0}^{\infty} n^2x^n$$
Then use that to find an expression for
$$\sum_{n=0}^{\infty}\frac{n^2}{2^n}$$
This is a... | there is a wrong sign in the first differentiation. I'd say:
$\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$
Differentiating (and multiplying with $x$)we have,
$\displaystyle \frac{x}{(1-x)^2}=\sum_{n=0}^{\infty}nx^n$
Differentiating(and multiplying with $x$) we have,
$\displaystyle \frac{x(1+x)}{(1-x)^3}=\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 0
} |
$x^2\dfrac{\partial u}{\partial x}+y^2\dfrac{\partial u}{\partial y}=u^2$ Help me please to solve the following PDE equation:
$x^2\dfrac{\partial u}{\partial x}+y^2\dfrac{\partial u}{\partial y}=u^2,\; \: u(x,2x)=1$
Thanks a lot!
| Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dx}{dt}=x^2$ , letting $x(1)=-1$ , we have $-\dfrac{1}{x}=t$
$\dfrac{dy}{dt}=y^2$ , we have $-\dfrac{1}{y}=t+y_0=-\dfrac{1}{x}+y_0$
$\dfrac{du}{dt}=u^2$ , we have $\dfrac{1}{u}=-t+f(y_0)=\dfrac{1}{x}+f\left(\dfrac{1}{x}-\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proof that $t^8+2t^6+4t^4+t^2+1$ is reducible in $\mathbb{F}_p$ Prove that the polynomial $t^8+2t^6+4t^4+t^2+1$ is reducible in $\Bbb F_p$, for all $p\in \Bbb P$.
Here are some examples:
$t^8+2t^6+4t^4+t^2+1=(1 + t + t^4)^2\pmod{2}$
$t^8+2t^6+4t^4+t^2+1=(1 + t) (2 + t) (1 + t^2) (2 + 2 t^2 + t^4)\pmod{3}$
$t^8+2t^6+4t^... | According to Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$
you should calculate galois group of this polynomial over $\mathbb Q$.
If you ask Magma:
P:=PolynomialAlgebra(Rationals()); f:=x^8+2*x^6+4*x^4+x^2+1; G:=GaloisGroup(f); print G;
then you have
Permutation g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
When are both fractions integers? The sum of absolute values of all real numbers $x$, such that both of the fractions $\displaystyle \frac{x^2+4x−17}{x^2−6x−5}$ and $\displaystyle \frac{1−x}{1+x}$ are integers, can be written as $\displaystyle \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. What is the v... | Hints:
$$\text{I}\;\;\;\;\;\;\;\frac{x^2+4x−17}{x^2−6x−5}=1+\frac{10x-12}{x^2-6x-5}=1+2\frac{5x-6}{(x-3)^2-14}$$
$$\text{II}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1-x}{1+x}=\frac{2}{1+x}-1$$
Look at II: the first fraction on the RHS must be an integer, so... The values you found with this sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$x^4-4y^4=z^2$ has no solution in positive integers $x$, $y$, $z$. How do I prove that the diophantine equation $x^4-4y^4=z^2$ has no solution in positive integers $x$, $y$, $z$.
| Let show that the equation in question is equivalent to another one:$x^4+y^4=z^2$, called as $\Gamma$-equation.
If $p$ and $q$ and $r$ satisfy $(\Gamma)$, then, upon setting $x=r$, $y=pq$, and $z=p^4-q^4$, we obtain a solution of $x^4=4y^4+z^2$. Conversely, if $x$, $y$, and $z$ satisfy the equation in question, then th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove $ax^2+bx+c=0$ has no rational roots if $a,b,c$ are odd If $a,b,c$ are odd, how can we prove that $ax^2+bx+c=0$ has no rational roots?
I was unable to proceed beyond this: Roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
and rational numbers are of the form $\frac pq$.
| Consider a quadratic equation of the form $a\cdot x^2 + b\cdot x + c = 0$. The only way, it can have rational roots IFF there exist two integers $\alpha$ and $\beta$ such that
$$\alpha \cdot \beta = a\cdot c\tag1$$
$$\alpha + \beta = b\tag2$$
$$
Explanation\left\{ \begin{align}
if\,\alpha\cdot \beta &= a\cdot c,\\
\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 11,
"answer_id": 0
} |
How many ways are there to distribute 5 balls into 3 boxes, under additional conditions? How many ways are there to distribute 5 balls into 3 boxes if:
*
*both the boxes and balls are labeled
*the balls are labeled but the boxes are not
*the balls are unlabeled but the boxes are labeled
*both the balls and boxes ... | (2) Since the boxes are indistinguishable, there are 5 different cases for arrangements of the number of balls in each box: $(5,0,0)$, $(4,1,0)$, $(3,2,0)$, $(3,1,1)$, or $(2,2,1)$.
$(5,0,0)$: There is only $1$ way to put all 5 balls in one box.
$(4,1,0)$: There are $\binom{5}{4} = 5$ choices for the 4 balls in one of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/340890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
find $n$ so $n/k$ is a $k$th power, $k=2,3,5$. Find a natural number n, in canonical form, such that:
$n/2=a^2$
$n/3=b^3$
$n/5= c^5$
for some a,b and c (natural numbers).
| Attempt an $n$ of the form $n = 2^x 3^y 5^z$.
You have $n/2 = 2^{x-1} 3^y 5^z$ as a square, so $x \equiv 1 (\bmod 2),\; y \equiv z \equiv 0 (\bmod 2).$
Analogously, $x \equiv z \equiv 0 (\bmod 3)$, $y \equiv 1 (\bmod 3)$ and
$x \equiv y \equiv 0 (\bmod 5)$, $z \equiv 1 (\bmod 5)$.
The Chinese remainder theorem guaran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/342712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$ $$\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$$
I have attempted this question by expanding the left side using the cosine sum and difference formulas and then multiplying, and then simplifying till I replicated the identity on the right. I am not s... | You can use the addition theorem which states that
$$\cos(\alpha+\beta)=\cos(\alpha)\cdot \cos(\beta) -\sin(\alpha)\sin(\beta)$$
$$\cos(\alpha + \beta) \cdot \cos(\alpha -\beta)= ( \cos(\alpha)\cos(\beta) -\sin(\alpha)\sin(\beta))\cdot (\cos(\alpha) \cdot \cos(-\beta)-\sin(\alpha)\cdot \sin(-\beta))$$
As $\cos(x)=\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/345703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 1
} |
Simplifying $\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt {5 +\cdots}}}}$ How to simplify the expression:
$$\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt{\cdots}}}}.$$
If I could at least know what kind of reference there is that would explain these type of expressions that would be very helpful.
Thank you.
| Let $x = 2\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt{...}}}}$. Then (if this converges) $x = 2\sqrt{5+x}$. Solving, $x = 2(1+\sqrt6)$, so the answer to your original question is $1+\sqrt{6}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/346303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 2
} |
How to prove $\cos\left(\pi\over7\right)-\cos\left({2\pi}\over7\right)+\cos\left({3\pi}\over7\right)=\cos\left({\pi}\over3 \right)$ Is there an easy way to prove the identity?
$$\cos \left ( \frac{\pi}{7} \right ) - \cos \left ( \frac{2\pi}{7} \right ) + \cos \left ( \frac{3\pi}{7} \right ) = \cos \left (\frac{\pi}{3}... | Hint
*
*$$ \cos A + \cos B = 2 \cos \left( \dfrac{A + B}{2} \right) \cos \left( \dfrac{A - B}{2} \right) $$
*$$ \cos A - \cos B = - 2 \sin \left( \dfrac{A + B}{2} \right) \cos \left( \dfrac{A - B}{2} \right) $$
*$$ \cos \left( \dfrac{2 \pi}{7} \right) = \cos { \left( 2 \theta \right) } \tag{ $ \theta = \dfrac{\pi}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
Proving $\tan \left(\frac{\pi }{4} - x\right) = \frac{{1 - \sin 2x}}{{\cos 2x}}$ How do I prove the identity:
$$\tan \left(\frac{\pi }{4} - x\right) = \frac{{1 - \sin 2x}}{{\cos 2x}}$$
Any common strategies on solving other identities would also be appreciated.
I chose to expand the left hand side of the equation and g... | As @ChristopherErnst suggests in a comment, some things become "more obvious after experience". Here are two alternative approaches to your problem that bear this out.
If you find yourself working with double- and half-angle arguments often, you might get the immediate sense that the right-hand side would be better if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Probability: Permutations
Consider the experiment of picking a random permutation $\pi$ on $\{1,2,...,n\}$, and define the associated random variable $f(\pi)$ as the number of fixed points of $\pi$, i.e, the number of $i$ such that $f(i)=i$.
I know that a permutation of $X=\{1,2,\ldots ,n\}$ is a one-to-one function ... | These can be done using generating functions. First, consider $E[F].$ The exponential generating function of the set of permutations by sets of cycles where fixed points are marked is
$$ G(z, u) = \exp\left(uz - z + \log \frac{1}{1-z}\right) =
\frac{1}{1-z} \exp(uz-z).$$
Now to get $E[F]$ compute
$$ \left.\frac{d}{du} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Integral solutions of hyperboloid $x^2+y^2-z^2=1$ Are there integral solutions to the equation $x^2+y^2-z^2=1$?
| The advanced approach is to rewrite this as $x^2+y^2= z^2+1$ and use unique factorization in $\mathbb Z[i]$. Assuming you disallow the obvious answers where $x$ or $y$ is $\pm 1$, you get that $x+yi=(a+bi)(c+di)$ and $z+i=(a+bi)(c-di)$ for some $a,b,c,d$. That yields a condition on $a,b,c,d$ and an explicit formulas f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/351491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 9,
"answer_id": 0
} |
How many solutions does $x^2 \equiv {-1} \pmod {365}$ have?
How many solutions does $x^2 \equiv {-1} \pmod {365}$ have?
My thought:
$365 = 5 \times 73$ where $5$ and $73$ are prime numbers.
So we can obtain $x^2 \equiv {-1} \pmod 5$ and $x^2 \equiv {-1} \pmod {73}$.
For $x^2 \equiv {-1} \pmod 5$,
we checked $5 \equ... | Working modulo $\,73\,$ :
$$2^6=-9\implies 2^{12}=(-9)^2=8=2^3\implies 2^9=1$$
$$3^4=8=2^3\implies3^{12}=(2^3)^3=2^9=1$$
$$5^6=3\implies 5^{36}=3^6=8\cdot3^2=72=-1$$
Thus, we've found a primitive root modulo $\,73\,$ (namely $\,5\,$), and from here we get two solutions to $\,x^2=-1\pmod{73}\,$ : $\;\;5^{18}=(5^6)^3=3^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/352585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Determining series formula Is there any procedure to follow when determining the function of a series? This seems simple but for I can't figure it out.
$$ \frac15 + \frac18 + \frac1{11} +\frac1{14} + \frac1{17}+\cdots$$
| Look at the denominators: $5, 8, 11, 14, 17, \ldots$ Can you find an expression for these?
Start by looking at the differences: $8-5=3$, $11-8=3$, $14-11=3$ and $17-14=3$. To get from one term to the next, we simply add $3$; the term-to-term rule is $+3$. That means that the sequence $5,8,11,14,17,\ldots$ is like the t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Triangle proof using law of sines
In triangle $ABC$, suppose that angle $C$ is twice angle $A$. Use the law of sines to show that $ab= c^2 - a^2$.
| Put $\,\angle C=2w\;,\;\;\angle A=w\;,\;\;\angle B=180- 3w\;$ , then
$$\frac{b}{\sin 3w}=\frac{c}{\sin 2w}=\frac{a}{\sin w}\implies ab=\frac{b^2\sin w}{\sin 3w}{}\;,\;\;a^2-c^2=\frac{b^2\sin^22w}{\sin^33w}-\frac{b^2\sin^2w}{\sin^23w}\implies$$
$$ab=c^2-a^2\iff \frac{b^2}{\sin 3w}\sin w=\frac{b^2}{\sin 3w}\left(\frac{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
How to factor $x^4 +3x -2$? I have figured out there is two roots between $0$ and $1 ,-1$ and $-2$ for $x^4 +3x -2 = 0$.
Therefore there should be two factors $(x + a)$ and $(y - b)$ where $a,b \in R^+$.
But how to find these $a$ and $b$?
When they found I can find the next factor in $ax^2+bx+c$ form and can check for... | For it to have some "nice" linear factors, the roots must be one of $\pm 1,\pm 2$ (this is due to the rational root theorem). You can quickly check that these are not the roots. The next bet is quadratic factors, i.e.,
$$(x^4+3x-2) = (x^2+ax+b)(x^2+cx+d)$$
Expanding the right hand side gives us
\begin{align}
a+c & = 0\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/356105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
What is the sum of all of the products of $3$ of the digits $1, \dots, 9$? Consider the numbers $1, 2, 3, \dots, 9$. Take the product of any three of them. What is the sum of all such products?
In other words, calculate $1 \cdot 2 \cdot3 + 1 \cdot 2 \cdot 4 + 1 \cdot 2 \cdot 5 + \dots + 7 \cdot 8 \cdot9$.
If we consi... | Use the Principle of Inclusion and exclusion.
Your sum is equal to
$$ \left[ (1 + 2 + \ldots + 9) ^3 - 3 \times ( 1^2 + 2^2 + \ldots + 9^2) \times (1 +2 + \ldots + 9 ) + 2 \times (1^3 + 2^3 + \ldots + 9^3) \right] \div 6$$
As an explanation, you can see if that $a\neq b, b\neq c, c\neq a$, then it will appear 6 times... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
The minimum of a function Could anyone possibly give me any help with finding the minimum of this function? I believe the result to be $2\pi |n|$ from page 619 of this paper by W. G. C. Boyd.
\begin{equation}
\frac{\frac{1}{2}(1+\zeta(m))\Gamma(m)}{(2\pi)^{m+1}|n|^m}
\end{equation}
Thanks!
| When $m$ is large, we have
$$
a_m : = \frac{{\frac{1}{2}\left( {1 + \zeta \left( m \right)} \right)\Gamma \left( m \right)}}{{\left( {2\pi } \right)^{m + 1} \left| n \right|^m }} \sim \frac{{\Gamma \left( m \right)}}{{\left( {2\pi } \right)^{m + 1} \left| n \right|^m }} \sim \frac{{m^{m - 1/2} e^{ - m} \sqrt {2\pi } }}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
"Incremental" hypergeometric distribution? Box contains $5$ yellow and $3$ red balls. $4$ balls are drawn without replacement. Let $X$ be the number of yellow balls appearing in the first two draws, and let $Y$ be the number of yellow balls appearing in total. Give the joint probability distribution of $X$ and $Y$.
| We want to find, for all relevant $x$ and $y$, the probability that $X=x$ and $Y=y$. Call this number $f_{X,Y}(x,y)$, or, for simplicity, $f(x,y)$.
There are not many possible values of $x$ and $y$, so we can find $f(x,y)$ for each combination. (Alternately, we can find a general formula for $f(x,y)$. For these small ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"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.