Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Proving that $a_n=\sqrt{n^2+n}-n$ converges, finding its limit and showing that its sequence ${(a_n)^{\infty}_{n=1}}$ is monotinic. As the title says, below I've proved the statement. Just posted here for verification and correction! And of course for other people to be inspired.
Let $a_n=\sqrt{n^2+n}-n$ with $n\in\mat... | A simpler approach: $a_n>0$,
$$a_n^{-1}=\frac{1}{\sqrt{n^2+n}-n}=\frac{\sqrt{n^2+n}+n}{n}=\sqrt{1+n^{-1}}+1$$
is decreasing and $a_n^{-1}\to 2$. So $(a_n)$ is increasing and $a_n\to 1/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
How can I find the primitive of $(x^3+x^2)/(4-x^2)$? I’m trying to find the primitive of the below rational function. I’m stuck trying to separate the undetermined coefficients like:
$$\frac{x^3+x^2}{(2-x)(2+x)}=\frac{A}{2-x}+\frac{B}{2+x}\iff x^3+x^2=A(2+x)+B(2-x)$$
What am I doing wrong?
| Hint :
$\dfrac{x^3+x^2}{(2-x)(2+x)}=-x-1+\dfrac{A}{2-x}+\dfrac{B}{2+x}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2545041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding the degree of minimal polynomial of a $10 \times 10$ matrix with entries $a_{ij}=1-(-1)^{i+j}$?
Q. What is the degree of minimal polynomial of a $10 \times 10$ matrix with entries $a_{ij}=1-(-1)^{i+j}$?
My approach : Let the matrix be denoted by $A$. Then $$A=
\left(\begin{matrix}
0 & 2 & 0 & 2 & 0 & ... | Check that $(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)$ is an eigenvector with eigenvalue $10$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
solution using synthetic geometry I managed to solve this problem only using complex numbers but I'd like to solve it using synthetic geometry and I can't.
Can someone help me to solve this problem using synthetic geometry?
Let $ABC$ an acute triangle with $AB > AC$ . Let $O$ its circumcenter and let $D$ the midpoint... | I was not able to solve it using synthetic geometry either. Is there anyone who can solve it and post a solution using synthetic geometry methods?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2550308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Let $\angle BAC =90 ^{\circ} AB=15 ,CD=10 ,AD=5$ Then $OA=?$
Let $\angle BAC =90 ^{\circ} AB=15 ,CD=10 ,AD=5$ Then $OA=?$
|
\begin{align}
x=|OA|&=|CO_1|=|A_1O_2|=|BO_3|
.
\end{align}
Let $|AD|=a$, $|DC|=2a$.
\begin{align}
|CO|&=2a\cos\theta
,\\
|AO|=|CO_1|&=
\sqrt{(3\,a)^2+(2a\,\cos\theta)^2-2\cdot3\,a\cdot(2\,a\,\cos\theta)
\cdot\cos\theta}
\\
&=a\,\sqrt{9-8\,\cos^2\theta}
,\\
|AO_1|&=\sqrt{|AC|^2-|CO_1|^2}
\\
&=\sqrt{(3a)^2-a^2\,(9-8\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Let $a^{4} + b^{4} + a^{2} b^{2} = 60 \ \ a,b \in \mathbb{R}$ Then prove that : Let $a^{4} + b^{4} + a^{2} b^{2} = 60 \ \ a,b \in \mathbb{R}$ Then prove that :
$$4a^{2} + 4b^{2} - ab \geq 30$$
My attempt: :
$$4a^{2} + 4b^{2} - ab \geq 30 \\ 4(a^2+b^2)-ab \geq30 \\4(60-a^2b^2)-ab\geq30\\ 240-30\geq4(ab)^2+ab\\ 4(ab)^2+... | $$a^4+b^4+a^2b^2=(a^2+b^2)^2-a^2b^2=60$$
$$(a^2+b^2-ab)(a^2+b^2+ab)=60$$
Solving equations $$a^2+b^2-ab=6$$$$a^2+b^2+ab=10$$ will give you the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2552804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Finding: $\lim\limits_{x\to -\infty} \frac{6x^2+5\cos{\pi x}}{\sqrt{x^4+\sin{5\pi x}}}$
I'm running into problems with this limit:
$$\lim_{x\to -\infty} \frac{6x^2+5\cos{\pi x}}{\sqrt{x^4+\sin{5\pi x}}}$$
I've tried using l'Hospitals rule, however we will alway keep the $\cos(\pi x)$ expression, as well for $\sin(5... | Easy using Squeeze theorem
$$ 6x^2-5\le6x^2+5\cos{\pi x}\le 6x^2+5$$
and
$$ 6x^4-1\le6x^2+\sin{5\pi x}\le 6x^4+1$$
then for $x<-1$ we have
$$\frac{6x^2-5}{\sqrt{x^4+1}}\le \frac{6x^2+5\cos{\pi x}}{\sqrt{x^4+\sin{5\pi x}}} \le \frac{6x^2+5}{\sqrt{x^4-1}}$$
By squeeze theorem we get
$$\lim_{x\to -\infty} \frac{6x^2+5\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2555213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Calculate the derivative using limit definition. This is the function $f(x)$$=\frac{1}{\sqrt{3x-2}}$ .
I wrote that $$\lim_{h\to 0}\frac{\frac{\sqrt{3x+3h-2}}{3x+3h-2}-\frac{\sqrt{3x-2}}{3x-2}}{h}.$$
I am not able to continue further.
| HINT:
Note that
$$\begin{align}
\frac{1}{\sqrt{3(x+h)-2}}-\frac{1}{\sqrt{3x-2}}&=\frac{\frac{1}{3(x+h)-2}-\frac{1}{3x-2}}{\frac{1}{\sqrt{3(x+h)-2}}+\frac{1}{\sqrt{3x-2}}}\\\\
&=\frac{\frac{-3h}{(3(x+h)-2)(3x-2)}}{\frac{1}{\sqrt{3(x+h)-2}}+\frac{1}{\sqrt{3x-2}}}
\end{align}$$
Divide by $h$ and let $h\to 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2556339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $x,y,z\in {\mathbb R}$, Solve this system equation: If $x,y,z\in {\mathbb R}$, Solve this system equation:
$$
\left\lbrace\begin{array}{ccccccl}
x^4 & + & y^2 & + & 4 & = & 5yz
\\[1mm]
y^{4} & + & z^{2} & + & 4 & = &5zx
\\[1mm]
z^{4} & + & x^{2} & + & 4 & = & 5xy
\end{array}\right.
$$
This is an olympi... | I checked the Possible Solutions thoroughly. The following solution is possible:
$$x^4+y^2+4+y^4+z^2+4+z^4+x^2+4-5yz-5xz-5xy=0 \Rightarrow (x^4-4x^2+4)+(y^4-4y^2+4)+(z^4-4z^2+4)+\left(\frac {5x^2}{2}-5xy+\frac {5y^2}{2} \right)+\left(\frac {5x^2}{2}-5xz+\frac {5z^2}{2} \right)+\left(\frac {5y^2}{2}-5yz+\frac {5z^2}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2556960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
Can we find the limit $\lim_{x\rightarrow\infty}\sum_{n=1}^{\infty}\frac{\left(-1\right)^nx^2}{n^2+x^2}$ without evaluating the sum? How to find the limit $\displaystyle\lim_{x\rightarrow\infty}\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}x^{2}}{n^{2}+x^{2}}$ if we don't evaluate the sum?
I know the sum is actually ... | Thanks for @Fimpellizieri , but I think I have got the answer.
We know $$\frac{x^2}{\left(2n\right)^{2}+x^{2}}-\frac{x^2}{\left(2n-1\right)^{2}+x^{2}}=\frac{\left(1-4n\right)x^2}{\left(\left(2n\right)^2+x^2\right)\left(\left(2n-1\right)^2+x^2\right)}$$
Consider
$$\left|\frac{-4nx^2}{\left(\left(2n\right)^2+x^2\right)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2557962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Transform complex equation into Cartesian coordinate Here is the problem
$$$$
Show the following equation is an ellipse
$$|z-1|=3-|z-2|$$
and I tried to solve it...
Square both sides,
$$(|z-1|)^2=(3-|z-2|)^2 \\
x^2+y^2-2x+1=9-6|z-2|+x^2+y^2-4x+4$$
Rearrange them...
$$6|z-2|=12-2x \\
3|z-2|=6-x$$
Square them again, and ... | So you get $8x^2+9y^2-24x=0$, which is $8\Big(x^2-3x+\frac{9}{4}\Big)-18+9y^2=0$, which is
$$8\Big(x-\frac{3}{2}\Big)^2+9y^2 = 18$$
which is an ellipse.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2561378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove the inequality $\frac{b+c}{a(y+z)}+\frac{c+a}{b(z+x)}+\frac{a+b}{c(x+y)}\geq \frac{3(a+b+c)}{ax+by+cz}$
Suppose that $a,b,c,x,y,z$ are all positive real numbers. Show that
$$\frac{b+c}{a(y+z)}+\frac{c+a}{b(z+x)}+\frac{a+b}{c(x+y)}\geq \frac{3(a+b+c)}{ax+by+cz}$$
Below are what I've done, which may be misleadi... | Too long for a comment.
Since the inequality is homogeneous, without loss of generality we may suppose $a+b+c=1$ and $x+y+z=1$. Then
$$(1-x)(1-y)(1-z)=1-x-y-z+xy+xz+yz-xyz=xy+xz+yz-xyz$$
Thus the left hand side of the inequality equals
$$\frac{1-a}{a(1-x)}+ \frac{1-b}{(1-y)}+ \frac{1-c}{c(1-z)}=$$
$$\frac{1}{a(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 0
} |
Solve the differential equation: $-x^2 y'' = \lambda y$ I need to solve the differential equation $-x^2 y'' = \lambda y$ by transforming the differential equation to a equation with constant coefficients. I need to do this by using $f(x) = y(e^x)$. If I do this, I become the equation:
$y''(e^x) + \frac{y'(e^x)}{e^x}+ ... | It is true that using $f(X)=y(e^X)$ allows to transform the ODE into an ODE with constant coefficients, but with a little trick at first beginning :
$f(X)=y(e^X)$ is valid any symbol of variable, for example $f(t)=y(e^t)$
Let $e^t=x \quad\to\quad f(t)=y(x)$
$dx=e^tdt=xdt \quad\to\quad \frac{dt}{dx}=\frac{1}{x}$
$\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Quadratic equation with different indices My maths teacher gave me a worksheet to work through as I was getting slightly bored in lessons. However, there was one question which I cannot do. The worksheet gives the answer, but you are supposed to show how you did it. Here is the question:
$729 + 3^{2x+1} = 4\times3^{x... | There was a typo in the question. It should have been
$$729+3^{2x+1}=4\times3^{x+3}$$
Note the $+3$ in the exponent instead of $+2$.
(You are correct that the question as stated does not have integer solutions, in fact it has no real solutions).
There's actually some pretty mathematical content here though. We can rewr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Integer solutions of $2x+3y=n$ What is the smallest $m$ such that for all $n\geq m$, the equation $2x+3y=n$ has solutions with $x,y \in \mathbb{Z}$ and $x,y\geq2$?
My approach. We can write the solutions in terms of the parameter $t$ as:
$$x(t)=-n-3t$$ and
$$y(t) = n+2t$$
Setting both of those greater than or equal to... | We consider two cases: $n$ is even and $n$ is odd.
If $n$ is even then $2x+3y = n \implies$ $y$ is an even number. The minimum possible value of even $n$ would be then $2(2)+3(2)=10$. For all even $n \ge 10$, we can set $x = n/2-3$ and $y=2$ to satisfy $2x+3y=n$.
If $n$ is odd then $2x+3y=n$ implies that $y$ is odd. C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Use of Taylor series Let $c>0$ a fixed parameter.
By using the Taylor's series I want to prove that there exists a constante $A>0$ which does not depend on $c$ such that $$\ln\left(1-\frac{2f(t)}{f(t)+g(t)}\right)\ge A(\frac{c+1}{4})t^3$$ for all $t\le \frac{1}{\sqrt c}$
where for all $t>0$:
\begin{align}
f(t) & =\fra... | I am not sure how much this could help you.
Using
$$f(t) =\frac{1-\cos(t\sqrt{c})}{c}-(\cosh(t)-1) $$
$$g(t) =\frac{\sin(t\sqrt{c})}{\sqrt{c}}+\sinh(t)$$ truncated Taylor series built around $t=0$ are
$$f(t)=-\frac{1}{24} (c+1) t^4+\frac{1}{720} \left(c^2-1\right)
t^6-\frac{\left(c^3+1\right)}{40320} t^8+O\left(t^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Sequence : $a_{n+1}=2a_n-a_{n-1}+2$ Let $c$ be a positive integer. The sequence $a_1, a_2, \ldots$ is defined by $a_1=1, a_2=c$ and
$a_{n+1}=2a_n-a_{n-1}+2$ for all $n \geq 2$.
Prove that for each $n \in \mathbb{N}$ there exists $k \in \mathbb{N}$ such that $a_na_{n+1} = a_k$.
My attempt :
Trying with small numbers,... | Since
$$
(a_{n+1}-a_n)-(a_n-a_{n-1})=2\tag1
$$
we have
$$
a_{n+1}-a_n=a_2-a_1+2n-2\tag2
$$
Thus,
$$
\begin{align}
a_{n+1}
&=a_1+(a_2-a_1)n+n^2-n\\
&=a_1+(a_2-a_1-1)n+n^2\tag3
\end{align}
$$
Using $(3)$, we can get by expansion
$$
a_na_{n+1}=a_{n^2+(a_2-a_1-2)n+2a_1-a_2+2}\tag4
$$
That is,
$$
\bbox[5px,border:2px solid ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Finding the maximum of $p^3 + q^3 +r^3 + 4pqr$
$p$,$q$,$r$ are $3$ non-negative real numbers less than or equal to $1.5$ such that $p+q+r = 3$, what will be the maximum of $p^3 + q^3 + r^3 + 4pqr$ ?
I tried AM-GM on $p,q,r$ to get the maximum of $pqr$ as $1$, but on doing it for $p^3 + q^3 + r^3 $, I get the minimum... | A hint before the answer:
$p^3+q^3+r^3+4pqr=p^3+(q+r)^3+4pqr-3(q+r)(qr)$
Try to then eliminate both $q$ and $r$ to make this in terms of $p$.
Motivation
Firstly one might try $p=q=r=1$ and get $p^3+q^3+r^3+4pqr=7$
But the trying of more cases will reveal that this is not the maximum. One might then try to fix a term a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Wrong Wolfram Alpha result for $\lim_{(x,y)\to(0,0)}\frac{xy^4}{x^4+x^2+y^4}$?
I'm trying to solve this limit:
$$
\lim_{(x,y)\to(0,0)}\frac{xy^4}{x^4+x^2+y^4}
$$
Here's my attempt:
$$0 \le |\frac{xy^4}{x^4+x^2+y^4} - 0| = \frac{|x|y^4}{x^4+x^2+y^4},$$ and since $x^4+x^2 \ge0$ then $\frac{y^4}{x^4+x^2+y^4} \le 1$ s... | Both of
$$\frac{xy^{4}}{x^{4}+x^{2}+y^{4}},\,\,\frac{xy^{4}}{x^{4}+x^{2}+y^{2}}$$
have limit $0.$ For the first one, you gave a valid proof. The second one follows from this one since the denominator is at least as big, while the numerator is the same.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Simplify $\frac{7^{ \log_{5} 15 }+3^{2+\log_{5}7}}{7^{\log_{5}3}}$ I know that the result of this expression is 16 but how do I get to that result?
$$\frac{7^{ \log_{5} 15 }+3^{2+\log_{5}7}}{7^{\log_{5}3}}$$
| Use properties of logs first:
\begin{align*}
\dfrac{7^{\log_5 15} + 3^{2+\log_5 7}}{7^{\log_5 3}}
= \dfrac{7^{\log_5 5 + \log_5 3} + 3^{2}3^{\log_5 7}}{7^{\log_5 3}}
&= \dfrac{7^{1}7^{\log_5 3} + 3^{2}3^{\log_5 7}}{7^{\log_5 3}} \\
&= 7 \biggl( \dfrac{7^{\log_5 3}}{7^{\log_5 3}}\biggr) + 9 \biggl(\dfrac{3^{\log_5 7}}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Why is function domain of fractions inside radicals not defined for lower values than those found by searching for domain of denominator in fraction? Consider function $y = \sqrt{\frac{1-2x}{2x+3}}$. To find the domain of this function we first find the domain of denominator in fraction:
$2x+3 \neq 0$
$2x \neq -3$
$x \... | ($1-2x\geq 0\text{ and }2x+3>0$) or ($1-2x\leq0\text{ and }2x+3<0$). From the first condition we get $x\in (\frac{-3}{2},\frac{1}{2}]$. Second condition isn't possible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
How to prove $\{n!e\} = \frac{1}{n +1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \dots$ I have the definition $a_n = \frac{1}{n +1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \dots$
I need to show that:
$a)$ $0 < a_n < \frac{1}{n}$
$b)$ $a_n = n!e - \lfloor{n!e}\rfloor$
So I know that
$\frac{1}{n} ... | By comparison with the Geometric series we have $$0<a_n<\frac 1{n+1}+\frac 1{(n+1)^2}+\cdots=\frac 1{n+1}\times \frac 1{1-\frac 1{n+1}}=\frac 1n$$
Now assume $n>1$ (so $0<a_n<\frac 1n<1$). We write: $$n!e=n!+ \frac {n!}{2!}+\cdots \frac {n!}{n!}+a_n$$ and it is clear that the part preceding $a_n$ is an integer (each... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2573192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
How I do evaluate this polynomial problem? Let $f(x)=x^2+ax+b$.
Suppose $f(f(x))=0$ equation has $4$ different real solutions $x_1,x_2,x_3,x_4$
and that two of them sum up to $-1$ (i.e. $\exists\, i\neq j$ such that $x_i+x_j=-1$)
Prove that $b\lt-\frac14$
| $f(x_k)$ are real roots of $f$, not all the same, so that $4b<-a^2$ is a necessary condition. If $z=f(x_2)=f(x_2)$ are the same root of $f$, then $x_1,x_2$ are the two solutions of $$f(x)=z\iff 0=x^2+ax+b-z$$ so that by Viete $a=-(x_1+x_2)=1$ and $b<-\frac14$ is necessary.
If $f(x_1)\ne f(x_2)$ then by Viete
\begin{al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2574097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
How to approach this minimization problem? I see this problem somewhere (it is from an olympiad or something but Im not sure where it comes):
Minimize $$\frac1{\sqrt{y+\frac1x+\frac12}}+\frac1{\sqrt{z+\frac1y+\frac12}}+\frac1{\sqrt{x+\frac1z+\frac12}}\tag1$$ for $x,y,z>0$ and $xyz=1$.
My first thought was approach th... | First, take $\displaystyle (x, y, z) = \left(t, t, \frac{1}{t^2}\right)$ and make $t \to +\infty$ to see that the infimum is no greater than $\sqrt{2}$. Next it will be proved that the infimum is $\sqrt{2}$.
Set $x = a^3$, $y = b^3$, $z = c^3$, then $abc = 1$ and\begin{align*}
(1) &= \frac{1}{\sqrt{\frac{b^3}{abc} + \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why is my alternate method of calculating scalar products not working? The exercise is such: Given that $|\vec{a}| = 3$, $|\vec{b}| = 2$ and $\varphi = 60^{\circ}$ (the angle between vectors $\vec{a}$ and $\vec{b}$), calcluate scalar product $(\vec{a}+2\vec{b}) \cdot (2\vec{a} - \vec{b})$.
My initial thought was to sol... | In the first method you assume at the end that the angle between $\vec{(a+2b)}$ and $\vec{(2a-b)}$ is $60^\circ$. If you were going to do this approach you would need the angle between them instead of $\cos 60^\circ$ for the last line.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
How to extend the matrix with determinant 1 to keep it Lets consider 2x2 integer matrix with determinant equal 1:
$$\left(
\begin{array}{cc}
a & b \\
c & d \\
\end{array}
\right)$$
I am working on the following:
How to extend this to 3x3 matrix in order to get another matrix with determinant 1:
$$\left(
\begin{array}... | You can just set $g = 1$ and $e,f,i,h = 0$:
$$1 = \begin{vmatrix}
a & b \\
c & d \\
\end{vmatrix} = \begin{vmatrix}
a & b & 0\\
c & d & 0\\
0 & 0 & 1
\end{vmatrix}$$
Furthermore, the extension is never unique since:
$$1 = \begin{vmatrix}
a & b \\
c & d \\
\end{vmatrix} = \begin{vmatrix}
a & b & 0\\
c & d & 0\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Find $(a,b)$ such that $(a^2+b)(a+b^2)=2^n$. Find all pair of positive integer numbers $(a,b)$ such that $(a^2+b)(a+b^2)=2^n$ for some positive integer number $n$.
Attempt I guest that there is only $(a,b)=(1,1)$ satisfying the assumption.
Thank you for all solution.
| We assume $a\geq b$. Let $a^2+b = 2^u, b+a^2 = 2^v$ with $u\geq v\geq 1$. Then
$$a^2+b \equiv a+b^2 \equiv 0 \pmod{2^v} \implies a^4+a \equiv b^4 + b \equiv 0 \pmod{2^v}$$
If $a,b$ are both odd, we have $$a^4 + a = a(a+1)(a^2-a+1) \equiv 0 \pmod{2^v} \implies a \equiv -1 \pmod{2^v}$$ similarly $b \equiv -1 \pmod{2^v}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find: $ \lim_{x\to \infty}\frac{x-1}{\sqrt{x}}-\frac{x-1}{\sqrt{x+1}}$
Find: $\displaystyle \lim_{x\to \infty} f(x)=\frac{x-1}{\sqrt{x}}-\frac{x-1}{\sqrt{x+1}}$
The answer provided in the book is 0 (also checked in Wolfram Alpha), but I can't find a good argument (without L'Hopital), to prove that. I end up in a $\in... | Note that $\frac{x-1}{\sqrt{x}} = \sqrt{x} - \frac{1}{\sqrt{x}}$, for example, so the limit is equal to $$\lim_{x \to \infty} \sqrt{x} - \sqrt{x+1} - \frac{1}{\sqrt{x}} + \frac{2}{\sqrt{x+1}}$$
which by arithmetic of limits is $$\lim_{x \to \infty} \sqrt{x} - \sqrt{x+1}$$
which is well-known to be $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 1
} |
A Question About A Calculation Of A Determinant.
Calculate $\det(A)$.
$$A =\begin{bmatrix}
a&b&c&d\\
-b&a&-d&c\\
-c&d&a&-b\\
-d&-c&b&a\\
\end{bmatrix}.$$
This is an answer on a book:
$$A A^T = (a^2+b^2+c^2+d^2) I.$$
$$\det(A) = \det(A^T).$$
$$\det(A)^2 = (a^2+b^2+c^2+d^2)^4.$$
The coefficient of $a^4$ in $\det(A)$ is... | why did I do like this? because above answer was formed by multiplying with transpose. if someone is interested in the direct solution, one can proceed as follows. It is too broad. I have learned about this technique today. I don't recommend this. Till $3\times 3$ this method is fine for this type of problems. For fun,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
If $(8+3\sqrt{7})^n = I + F $ where $ I $ is integer and $F $ is proper fraction, what is $I$? I am beginner to Binomial Theorem and I want to find out weather $I$ is even or odd in $$(8+3\sqrt{7})^n= I+F$$ if it can be expressed as a sum of an Integer $I$ and a proper fraction $F$
How could I find out ?
| Notice that
$$\begin{align}
(8 +3 \sqrt{7})^n + (8 - 3 \sqrt{7})^n &= \sum_{i=0}^n \binom{n}{i} 8 ^i (3 \sqrt{7})^{n-i} + \sum_{i=0}^n \binom{n}{i} (-1)^i 8 ^i (3 \sqrt{7})^{n-i} \\
&= 2 \sum_{i=0}^{\lfloor n/2 \rfloor} \binom{n}{2i} 8^{2i} 3^{2(n-i)} 7^{n-i} \qquad \text{(the odd numbered terms cancel out)}\\
&= 2 N
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the equations of the straight line through $(0,a)$ on which the perpendiculars dropped from the point $(2a,2a)$ are each of length $a$ unit. Find the equations of two straight lines drawn through the point $(0,a)$ on which the perpendiculars drawn from the point $(2a,2a)$ are each of length $a$.
My Attempt:
The eq... | Solution,
Given,
Points: (0,a) and (2a, 2a)
Now,
y - y1 = m (x - x1)
or, y - a = m (x - 0)
or, mx - y + a = 0.........(i)
Since, perpendicular distance of (i) from (2a, 2a) is a.
a = +- 2am - 2a + a / (root under) m ^ 2 + (-1) ^ 2
or, a (root under) m ^ 2 + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2595752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Can $10\uparrow^n m<2\uparrow^n (m+2)$ be formally proven? See here :
http://googology.wikia.com/wiki/Arrow_notation
for the definition of the up-arrow function.
Can $10\uparrow^n m<2\uparrow^n (m+2)$ be formally proven for all $m\ge 1$ and $n\ge 3$ ?
With Saibians theorem we get $$10\uparrow^n m<(2\uparrow^n 3)\upa... | For $n=3$, we have:
For $m=1$, we have $10\uparrow^31=10$ and $2\uparrow^33=65536$.
Assume $10\uparrow^3m<2\uparrow^3(m+2)-3$ holds for some $m\ge1$. Then we have
\begin{align}10\uparrow^3(m+1)&=10\uparrow^210\uparrow^3m\\&<10\uparrow^2(2\uparrow^3(m+2)-3)\\&<(2\uparrow^23)\uparrow^2(2\uparrow^3(m+2)-3)-3\tag0\\&<2\upa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Product of length of segments in Ellipse.
If the normal at any point $P$ on the Ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2} {b^2}=1$ meets the axis in $G$ and $g$ respectively, then find $PG\cdot Pg$, in terms of $a$ and $b$.
I tried considering the parametric point as $(a\cos\theta,b\sin\theta)$ on the ellipse, then cons... | Hint:
Using this, the equation of normal
$$\dfrac x{\dfrac{(a^2-b^2)\cos\theta}a}-\dfrac x{\dfrac{(a^2-b^2)\sin\theta}b}=1$$
$$|PG|=\sqrt{\left(\dfrac{(a^2-b^2)\cos\theta}a-a\cos\theta\right)^2+(0-b\sin\theta)^2}=?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2597836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Newton binomial expansion Find the coefficient for ${x^{11}}$ in ${\sqrt{1+x}}$
The generic formula is
$$\sqrt{1+x}=(1+x)^\frac{1}{2}=1+\frac{1}{2}x+\frac{\frac{1}{2}(\frac{1}{2}-1)}{2}x^2\dots$$
Solution of expansion of coeficient for ${x^{11}}$:
$${\frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)(\frac{1}{2}-3)(\frac{... | $(1+x)^a=1+ax+...+\dfrac{a(a-1)\cdots(a-(n-1))}{n!}x^n+o(x^n)$
Let show by induction $\sqrt{1+x}=\sum\limits_{k=0}^n a_kx^k+o(x^n)$ with $$a_n=\dfrac{(-1)^{n-1}}{(2n-1)}\dfrac{\binom{2n}{n}}{4^n}$$
$a_0=\dfrac{-1}{-1}=0\quad\checkmark$
$a_1=\dfrac{2}{4}=\frac 12\quad\checkmark$
$a_{n+1}=a_n\times\dfrac{(\frac 12-n)}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2603383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simple exersice using CRT Calculate all solutions $0\leq x< 130$ for the following system of equations.
$x \equiv_2 1$
$x \equiv_5 2$
$x \equiv_{13} 3$
Solution:
$M=2\cdot 5\cdot 13=130$
So by CRT there exists a unique solution.
We get:
$M_1=M/2=65$
$M_2 = M/5 = 26$
$M_3=M/13=10$
Now:
$65\cdot N_1 \equiv_2 1 \ \ \Righ... | If you set
$$26\cdot N_2 \equiv_5 1 \ \ \Rightarrow \ \ N_2=1 \\
10\cdot N_3 \equiv_{13} 1 \ \ \Rightarrow \ \ N_3=4
$$
Then you'll get the right number
$$x = R_{130}(1\cdot M_1 \cdot N_1 + 2\cdot M_2 \cdot N_2 + 3 \cdot M_3 \cdot N_3)\\=R_{130}(1 \cdot 65\cdot 1 + 2\cdot 26 \cdot 2 + 3 \cdot 10 \cdot 12)\\
=R_{130}(65... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2605035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to compute $\int_{0}^{1}\left (\frac{\arctan x}{1+(x+\frac{1}{x})\arctan x}\right )^2dx$ I want to calculate the following integral $$\int_{0}^{1}\left(\frac{\arctan x}{1+(x+\frac{1}{x})\arctan x}\right)^2 \, dx$$
But I have no way to do it, can someone help me, thank you.
| Here is a way to arrive at the answer, and it is by no means obvious.
In finding the indefinite integral the reverse quotient rule will be used. Recall that if $u$ and $v$ are differentiable functions, from the quotient rule
$$\left (\frac{u}{v} \right )' = \frac{u' v - v' u}{v^2},$$
it is immediate that
$$\int \frac{u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2607796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Find the ratio of segments using Ceva's theorem I am going through $IB$ further math geometry topic and just learned Ceva's theorem. Below is one of the questions in the exercise. I have thought for quite a while and cannot solve it. Can anyone give me some clues or hints, please? Thanks.
In the diagram, $BZ:ZC=2:1$ an... | The proof of Ceva'sTheorem uses the fact that if $ A, B $ and $ C $ are three non-collinear points then any vector $ P $ can be expressed as $ P = xA + yB + zC $ where $ x+ y + z = 1 $. Also, a point on the line joining $ A $ and $ B $ is given by $ P = tA + (1-t)B $.
$ Z = \frac {2}{3} C + \frac {1}{3} B $
$ S = \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Calculate limit of $(1 + x^2 + y^2)^\frac{1}{x^2 + y^2 + xy^2}$ as $(x,y) \rightarrow (0,0)$ I'm trying to calculate the limit of
$(1 + x^2 + y^2)^\frac{1}{x^2 + y^2 + xy^2}$ as $(x,y) \rightarrow (0,0)$.
I know that the limit is supposed to be $e$, and I can arrive at this answer if I study the univariate limit by, f... |
I thought it might be instructive to present an approach that circumvents use of polar coordinates and relies instead on a straightforward application of the AM-GM inequality. To that end we proceed.
To begin, the AM-GM inequality guarantees that for $x>-1$
$$\begin{align}
\left|\frac{xy^2}{x^2+(1+x)y^2}\right|&\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2610870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Question on complex numbers and cube root of unity
If $a$, $b$, $c$ are distinct integers and $\omega$ is a cube root of unity then minimum value of $|a + b\omega + c\omega^2| + |a+b\omega^2 + c\omega|$ is?
Now, I know the identity $a^3 + b^3 +c^3 -3abc=(a+b+c)(a + b\omega + c\omega^2)(a+b\omega^2 + c\omega)$.
I don... | Let
$$A=|a + b\omega + c\omega^2|$$
$$B=|a+b\omega^2 + c\omega|$$
note that $B=\bar A=A\implies A+B=2A$
then
$$A+B=2A=2\left|a + b\left(-\frac12+i\frac{\sqrt3}{2} \right) + c\left(-\frac12-i\frac{\sqrt3}{2} \right)\right|=2\left|a -\frac12b-\frac12c+i\frac{\sqrt3}{2}\left(b-c\right)\right|=2\left(a^2+\frac14b^2+\frac14... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that if $x+y+z=x^2+y^2+z^2=x^3+y^3+z^3=1$ then $xyz=0$ For any real numbers $x,y,z$ such that $x+y+z = x^2+y^2+z^2 = x^3+y^3+z^3 =1\\$ show that $x \cdot y \cdot z=0$.
I think that because $x \cdot y \cdot z = 0$ at least one of the three number should be equal to zero, but I'm stuck into relating the other things... | \begin{align}
\left( \sum_{cyc} x \right) \left( \sum_{\text{cyc}} x^2
\right) &= \left( \sum_{\text{cyc}} x^3 \right) = 1 \\
\sum_{cyc} x^2y + \sum_{cyc} xy^2 &= 0
\end{align}
In the above equation, subtract $x^3+y^3+z^3$ from both sides.
\begin{align}
(x+y+z)^3 &= 1 \\
\sum_{cyc} x^3 + 3\sum_{cyc} x^2y + 3\sum_{cyc} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 7
} |
Does $x+\sqrt{x}$ ever round to a perfect square, given $x\in \mathbb{N}$? I'll define rounding as $$R(x)=\begin{cases} \lfloor x \rfloor, & x-\lfloor x \rfloor <0.5 \\ \lceil x \rceil, & else\end{cases}$$
Does $x+\sqrt{x}$ ever round (to the nearest integer) to a perfect square, given $x\in \mathbb{N}$?
For example... | Assume $x\in\Bbb Z^+$.
If $x=m^2$ is a pefect square, then
$$m^2<x+\sqrt x=m^2+m<m^2+2m+1=(m+1)^2$$
an so $x+\sqrt x=R(x+\sqrt x)$ cannot be a perfect square.
Thus we need only consider the case that $x$ is not a perfect square, which makes $\lfloor x+\sqrt x\rfloor <\lceil x+\sqrt x\rceil$.
Let $n\in\Bbb Z^+$ be maxim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
How do I find all integers that satisfy this condition? Suppose that $72x + 56y = 40$. Find all $x,y$ that satisfy this condition.
Here is what I did:
*
*I reduced the equation to give $9x + 7y = 5$
*But since 5 is relatively prime to 7 and 9, we know $x,y$ is divisible by 5. So we can reduce the equation to $9(5a)... | $$9x+7y=5$$
First, you find a solution.
\begin{array}{rcll}
5-7(1) &= &-2 \\
5-7(2) &= &-9 \\
9(-1) + 7(2) &= &5
\end{array}
Because $\gcd(9,7)=1$, the general solution is
$$(x,y) = (-1+7t, 2-9t)$$
for all $t \in \mathbb Z$.
Now take the general case $9x+7y=5$ and subtract $9(-1)+7(2)=5$.
\begin{array}{cccc}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Sequence Convergence of $(s_n) = (\frac{1}{n^2})$ Part 1. Given $\varepsilon = 0.5$, find an $N \in \mathbb{N}$ s.t whenever $n \geq N$, $|s_n - 0| < \varepsilon$
&&
Part 2. Treat the epsilon in Part 1 with an arbitrary $\varepsilon > 0$
My work:
Let $s_n = \frac{1}{n^2}$, and suppose $s = 0$, implies $|\frac{1}{n^2} ... | Just use the ceiling function, we just have to choose an $N \in \mathbb{N}$ such that $N> \frac{1}{\sqrt{\epsilon}}$, you can choose it to be $1+\lceil \frac{1}{\sqrt{\epsilon}}\rceil.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
help with sum of infinite series, stuck in problem the series in the problem is as follows, and we would like to see if the series converges, and what is the sum of infinite series:
$$\sum_{n=1}^{\infty} \frac{2^{n+1}+1}{3^n}$$
It's a homework problem, but I already asked my teacher about it at school and he didn't rea... | After 20 years, I still have to rederive the identity
$$ \sum_{n=1}^{\infty} \alpha^n = \frac{\alpha}{1-\alpha} \tag{1}$$
whenever I need it. Since it isn't that difficult, perhaps we should do that first.
Suppose that $|\alpha| < 1$, and let
$$ S_k := \sum_{n=1}^{k} \alpha^k = \alpha + \alpha^2 + \dotsb + \alpha^{k} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2620771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Computing the definite integral $\int _0^a \:x \sqrt{x^2+a^2} \,\mathrm d x$
Compute the following definite integral $$\int _0^a \:x \sqrt{x^2+a^2} \,\mathrm d x$$
This is what I did:
$u = x^2 + a^2 $
$du/dx = 2x$
$du = 2xdx$
$1/2 du = x dx$
$\int _0^a\:\frac{1}{2}\sqrt{u}du = \frac{1}{2}\cdot \frac{u^{\frac{3}{2}}}{... | You can simply use the reverse chain rule for integration and you get:
$$\int_0^a x\sqrt{x^2+a^2}dx=\frac 12\int_0^a 2x\sqrt{x^2+a^2}dx=\left[\frac{(x^2+a^2)^\frac 32}3\right]_0^a=\\=\frac{(2a^2)^\frac 32}3-\frac{(a^2)^\frac 32}3=\color{red}{\frac{(2\sqrt 2-1)}3|a|^3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2621750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Partial Fraction problem solution deviates from the Rule Question:
Compute $\displaystyle \int\frac{x^2+1}{(x^2+2)(x+1)} \, dx$
My Approach:
As per my knowledge this integral can be divided in partial Fraction of form $\dfrac{Ax+B}{x^2+px+q}$ and then do the following as per to integrate it.
Solution:
Taking $\dfrac{x^... | From what I understand, the solution in your book points that the fraction can be writeen as
$$ \frac{x^{2}+1}{(x^{2}+2)(x+1)} = \frac{Ax^{2}+Bx+C}{x^{2}+2} + \frac{D}{x+1} $$
$$ = \frac{(Ax^{2}+Bx+C)(x+1) + D(x^{2}+2)}{(x^{2}+2)(x+1)} $$
Of course it can, you just need to find the appropriate constants :
$$ Ax^{3} +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2622263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Subgroups of General Linear Group I have a question about the order of the subgroups of $GL_2(\mathbb{R})$ generated by the following matrix:
$\begin{pmatrix}
1&-1 \\
-1&0
\end{pmatrix}$
So I computed several powers of this matrix and realized that the order of the subgroup generated by this matrix is infinity. Ho... | Hint: if $\lambda$ is an eigenvalue of a matrix $A$ (that is, there exists a nonzero column vector $v$ such that $Av = \lambda v$), then $\lambda^n$ is an eigenvalue of the matrix $A^n$.
Suppose that $A^n = I$ for some $n$. What are the eigenvalues of $I$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2626045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Prove the following statements without using induction The first statement:
$$\tan(nx)=\frac{C^n_1\tan(x)-C^n_3\tan^3(x)+C^n_5\tan^5(x)-C^n_7\tan^7(x)+\dotsm}{1-C^n_2\tan^2(x)+C^n_4\tan^4(x)-C^n_6\tan^6(x)+\dotsm}$$
where
$$C^n_k=\frac{n!}{k!(n-k)!}$$
The second statement:
$$\sin(x)\sin\Bigl(x+\frac{\pi}{n}\Bigr)\si... | Use Binomial & DeMoivre's Theorems
\begin{eqnarray*}
\cos(nx) &=& \operatorname{Re}(( \cos x+ i \sin x)^n) =\cos^n x - \binom{n}{2} \cos^{n-2} x \sin^2 x + \binom{n}{4} \cos^{n-4} x \sin^4 x - \cdots\\
\sin(nx) &=&\operatorname{Im}(( \cos x+ i \sin x)^n) =\binom{n}{1}\cos^{n-1} x \sin x- \binom{n}{3} \cos^{n-3} x \sin^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Beckenbach Introduction to Inequalities Chapter 2: Show $(a+b)/2 \le ( (a^2 + b^2 )/2)^{1/2}$ I'm having trouble understanding the following problem
Problem
Beckenbach, Chapter 2 Pg 24 Ex 1
$$
\text{Show the following for all a, b}\quad
\frac{(a+b)}{2} \le \left(\frac{a^2 + b^2}{2}\right)^\frac{1}{2}
$$
The book pr... | If left side is not positive then inequality obviously hold.
So assume it is nonegative. Square it: $${a^2+2ab+b^2\over 4}\leq {a^2+b^2\over 2}$$
Get rid of denumerators and you get
$$ a^2+2ab+b^2\leq 2a^2+2b^2$$
or $(a-b)^2\geq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to prove $\tan\Big[\frac{1}{2}\sin^{-1}\frac{3}{4}\Big]=\frac{4-\sqrt{7}}{3}$ Prove
$$
\tan\Big[\frac{1}{2}\sin^{-1}\frac{3}{4}\Big]=\frac{4-\sqrt{7}}{3}
$$
and justify why $\frac{4+\sqrt{7}}{3}$ is ignored.
My Attempt:
$$
\tan x=\frac{2\tan\frac{x}{2}}{1-\tan^2\frac{x}{2}}\implies\tan x-\tan x\tan^2\frac{x}{2}... |
Here's a nice (if I say so myself) geometric solution. Consider the right triangle $\triangle ABC$ with right $\angle ABC$, $|BC| = 3$ and $|AC| = 4$. $AD$ is the angle bisector of $\angle BAC$. Let $|BD| = x$ and hence $|CD| = 3-x$.
Now $\displaystyle |AB| = \sqrt{4^2 - 3^2} = \sqrt 7$ (Pythagoras). Note that $\displ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2632633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
How to obtain the right-hand side of the following equation? How to obtain the right-hand side of the following equation\begin{align*}
&\mathrel{\phantom{=}}\sum_{m=1}^{w-2}\sum_{n=2}^{2w-2m-1} 2^{n-2} g(2m+n,2w-2m-n)\\
&=\frac{1}{6}\sum_{k=4}^{2w-1}[2^{k-1}-3+(-1)^k] g(k,2w-k)
\end{align*}
by substituting $k=2m+n$ int... | There might be an easier way, but this is what I could do. First of all, I will ignore the function $g$, since it is clear that it does not have to do with what we want to show, namely:
$$ \sum_{m=1}^{w-2}\sum_{n=2}^{2w-2m-1}2^{n-2} = \frac{1}{6} \sum_{k=4}^{2w-1} \left(2^{k-1}-3+(-1)^k \right)$$
We start from the LHS.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2635357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the no. of ways to color the matrix.. Given a $3 \times N$ matrix and three colors $a$, $b$ and $c$. In how many ways can all the cells be colored such that no row and no column contains all the cells of same color ?
I could not reach to any logical approach to the problem .
| For $i=1,2,3$ let $R_i$ denote the set of colourings such that the cells in row $i$ get the same color.
For $j=1,\dots,N$ let $C_j$ denote the set of colourings such that the cells in column $j$ get the same color.
Then to be found is: $$3^{3N}-|R_1\cup R_2\cup R_3\cup C_1\cup\cdots\cup C_N|\tag1$$
For this we use in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2635488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Mnemonics for typical integrations I am supposed to mug up these integrals for my upcoming exams:
$$\int\sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac xa+C$$
$$\int\sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln|x+\sqrt{x^2+a^2}|+C$$
$$\int\sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\... | As bames suggested above, the method of solving for these indefinite integrations is actually a helpful tool in quickly recalling them. Here, I will provide a simple derivation - for the three integrations I listed - that helped me recall them within thirty seconds on paper. This is a long post, because it captures the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2636865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Some implicit differentiation questions. Please check them [see desc.] I have a few implicit diffentiatial questions that I wanted to check. general question... how do you know that $y$ is a function of $x$? I assume the whole reason why these are called implicit differentiation questions is because $y$ is defined impl... | 1) As John pointed there is a mistake...
2) The second example seems correct to me
3) For the third differenciation...
$$y \cdot \cos{x} = x^2 + y^2$$
$$y'\cos(x)-y\sin(x)=2x+2yy'$$
$$y'\cos(x)-2yy'=y\sin(x)+2x$$
$$y'(\cos(x)-2y)=y\sin(x)+2x$$
$$\frac {dy}{dx}=\frac {y\sin(x)+2x}{\cos(x)-2y} $$
...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2637866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How do I find the radius of convergence for this power series? I stumbled on this power series and was wondering how could I evaluate the interval of convergence:
$$\sum_{n=1}^{\infty }(-1)^nx^{3n}\frac{1\cdot 4\cdot 7\cdot ...\cdot (3n-2)}{n!\cdot 3^{n}}$$
I first tried using the ratio test to determine the $x$ interv... | You misunderstood the notation when you wrote down the "$n+1$" term. The ratio should be
$$x^{3(n+1)}\frac{1\cdot4\cdot7\cdots(3n-2)\cdot(3(n+1)-2)}{(n+1)!\, 3^{n+1}}
\bigg/x^{3n} \frac{1\cdot 4\cdot 7\cdots (3n-2)}{n!\,3^{n}}$$
which simplifies to
$$\frac{(3n+1)x^3}{3(n+1)}\ ,$$
and I think you will find it easy en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2638116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove the positivity of a sequence Let $a>0$ a real number and $(u_n)$ the sequence defined by
$$
u_{n+1} = a - \frac{1}{u_n}\text{ and } u_0 = a.
$$
Question: Determine condition on the value of $a>0$ such that the sequence $(u_n)$ is always positive.
Attempt: I tried to establish a general formula of $u_n$ in order... | Notice that if the sequence $u_n$ is positive, then it must be decreasing. This can be proven by induction.
In this case, the sequence $u_n$ is monotonic and bounded, so it has a limit $L$ that has to be positive.
From the formula,
$$L=a-\frac{1}{L}$$
Thus
$$a=L+\frac{1}{L} \ge 2$$
So a necessary condition is $a \ge 2$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2638855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Want to understand how a fraction is simplified The fraction is used to determine the sum of a telescopic series.
$$\sum_{k=1}^\infty\frac{1}{(k+1)\sqrt{k}+k\sqrt{k+1}}$$
This is the solved fraction.
$$\frac{1}{(k+1)\sqrt{k}+k\sqrt{k+1}} = \frac{(k+1)\sqrt{k}-k\sqrt{k+1}}{(k+1)^2k-k^2(k+1)} = \frac{(k+1)\sqrt{k}-k\sqrt... | Another way to look at it is first "declutter" the expression, since all those radicals obfuscate the simple structure. Let $a=\sqrt{k}\,$, $b=\sqrt{k+1}\,$, then:
$$\frac{1}{(k+1)\sqrt{k}+k\sqrt{k+1}} = \frac{1}{ab^2+a^2b}=\frac{1}{ab(a+b)}$$
Now consider that $\require{cancel}\,(b+a)(b-a)=b^2-a^2=(\cancel{k}+1)-\canc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How would one show that the integral $\int_0^\infty \frac 1 {(1+x)\sqrt x}\,\mathrm d x$ either diverges or converges? I'm supposed to be showing that the integral
\begin{equation}
\int_0^\infty \frac 1 {(1+x)\sqrt x}\,\mathrm d x
\end{equation}
either converges or diverges, using the comparison test.
It can easily be ... | By letting $y=\sqrt{x}$, then
\begin{align*}
\int_0^\infty \frac 1 {(1+x)\sqrt{x}}\,dx=\int_0^\infty\frac{2}{1+y^2}\,dy,
\end{align*}
and
\begin{align*}
\int_0^\infty \frac{1}{1+y^2} \, dy = \lim_{M\rightarrow\infty} \tan^{-1} y \bigg|_{y=0}^{y=M} = \frac\pi 2<\infty.
\end{align*}
Maybe some sort of comparison:
\begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
When is $(p^2-1)/8$ even? I am trying to find for what values of p $\frac{p^2-1}{8}$ will be even number $\frac{p^2-1}{8}=2m\implies (p-1)(p+1)=8(2m)$ then can I write $p\equiv 1\pmod8$ or $p\equiv -1\pmod8$ ?
Also trying to find for what values of p it will be an odd number
$\frac{p^2-1}{8}=(2m+1)\\\implies ... | $\Box \ 4\mid p^2 - 1$ for all odd $p$.
Proof: For some $n\in\mathbb{Z}$, since $p$ is odd, then $p = 2n+1$. $$\begin{align} \therefore 4\mid (2n + 1)^2 - 1 &= (2n)^2 + 1^2 + 4n - 1\tag1 \\ &= 4n^2 + 4n \\ &= 4(2n^2 + 2n).\tag*{$\Box$}\end{align}$$ Notice though that $2n^2 + 2n = 2(n^2 + n)$. $$\therefore 4\times 2 = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2645398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculate the determinant $\left|\begin{smallmatrix} a&b&c&d\\ b&a&d&c\\ c&d&a&b\\d&c&b&a\end{smallmatrix}\right|$ Question: Calculate the following determinant
$$A=\begin{vmatrix} a&b&c&d\\ b&a&d&c\\ c&d&a&b\\d&c&b&a\end{vmatrix}$$
Progress: So I apply $R1'=R1+R2+R3+R4$ and get
$$A=(a+b+c+d)\begin{vmatrix} 1&1&1&1\\ b... | \begin{align}A&=(a+b+c+d)\begin{vmatrix}a-b&d-b&c-b\\d-c&a-c&b-c\\c-d&b-d&a-d \end{vmatrix} \\&=(a+b+c+d)\begin{vmatrix}a-b&d-b&c-b\\d-c&a-c&b-c\\0 &a+b-c-d&a+b-c-d \end{vmatrix} \\
&= (a+b+c+d)(a+b-c-d)\begin{vmatrix}a-b&d-b&c-b\\d-c&a-c&b-c\\0 &1&1\end{vmatrix} \\
&= (a+b+c+d)(a+b-c-d)\begin{vmatrix}a-b&d-b&c-b\\a+d-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2647193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Identify $\mathbb{Z}[x]/(2x^2+1,2x-3)$. Identify $\mathbb{Z}[x]/(2x^2+1,2x-3)$.
I tried this:
$$
\mathbb{Z}[x]/(2x^2+1,2x-3)=:R \\
(2x^2+1,2x-3)=:I \\
\because (2x^2+1)-x(2x-3)=3x+1, \\
2(3x+1)-3(2x-3)=11 \\
\therefore I=(2x^2+1,2x-3,11) \\
\therefore R \cong \mathbb{Z}_{11}[x]/(2x^2+1,2x-3) \\
\cong \mathbb{Z}_{11}[x... | $2x-3=0 \implies x=\frac 3 2$. Substituting $\frac {11} 2=0$, thus of course $11=0$. Our ring becomes $\mathbb Z_{11}[\frac 3 2]$. Now we have to check that we already have an element in $\mathbb Z_{11}$ such that multiplying it by $2$ it becomes $3$: $$2y \equiv 3 (mod \hspace{0,2cm}11)$$ we multply by $6$ and we get ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2648387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
VERIFICATION: Apply Step-deviation method to find the average mean of the following frequency distribution Q: Apply Step-deviation method to find the average mean of the following frequency distribution
\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|} \hline
Variate (x_i) & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 \\ \hl... | The source of error is the wrong combination of the $2i-1$ and $2i$-th data into one class of wrong width $h=10$. You're in fact calculating
$$\sum_{i=1}^5 x_{2i}(f_{2i-1}+f_{2i}) \tag{overestimated}$$
instead of
$$\bar x = \sum_{i=1}^5 (x_{2i-1}f_{2i-1} + x_{2i}f_{2i}) = \sum_{i=1}^{10} x_i f_i.$$
Since the common di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2648816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Trig Subs issues Problem:
Use a trigonometric substitution to find
$$I = \int_\sqrt{3}^2{\frac{\sqrt{x^2-3}}{x}}\;dx$$
Give both an exact answer (involving $\pi$ and a square root) and a decimal estimate to 3 significant digits.
Here is the work I have so far.
$$I = \int_\sqrt{3}^2{\frac{\sqrt{x^2-3}}{x}}\;dx$$
$$1 + ... | You did well.
Swith the limit when you replace $dx$ by $d\Theta$.
\begin{align}
I&= \int_0^{\frac{\pi}6} \frac{\tan \Theta}{\sec \Theta} \sqrt{3} \tan \Theta \sec \Theta \, d\Theta \\
&= \sqrt{3} \int_0^{\frac{\pi}{6}} \tan^2\Theta\, d\Theta\\
&= \sqrt{3} \int_0^{\frac{\pi}6} \sec^2 \Theta -1 \,d\Theta\\
&= \sqrt{3} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2649804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $(x,y) \in \Bbb R^+$ Prove that $\Bigr(1+\frac{1}{x}\Bigl)\Bigr(1+\frac{1}{y}\Bigl)\ge \Bigr(1+\frac{2}{x+y}\Bigl)^2$ Let $(x,y) \in \Bbb R^+$ Prove that $$\Bigr(1+\frac{1}{x}\Bigl)\Bigr(1+\frac{1}{y}\Bigl)\ge \Bigr(1+\frac{2}{x+y}\Bigl)^2$$
My try
Well, i didn't see a way to factorize this, so i put it in WolframA... | It's just Jensen for $f(x)=\ln\left(1+\frac{1}{x}\right)$.
Indeed, $f''(x)=\frac{(2x+1)}{x^2(x+1)^2}>0$ and we get
$$\frac{\ln\left(1+\frac{1}{x}\right)+\ln\left(1+\frac{1}{x}\right)}{2}\geq\ln\left(1+\frac{1}{\frac{x+y}{2}}\right),$$
which is your inequality.
Also, it's
$$1+\frac{1}{x}+\frac{1}{y}+\frac{1}{xy}\geq1+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 2
} |
Sangaku Circle Geometry Problem I'm having difficulties with this Sangaku problem and was hoping for some help!
Five circles (1 of radius c, 2 of radius b, and 2 of radius a) are inscribed in a segment of a larger circle (note: this segment does not have to be a semi-circle). Given a and b, find c. For example, if a =... | Got the answer... Brutally... Took about 1 hour work and a page of letter size by hand. It at least seemed to me that using inversion did not make things simpler... But maybe someone can work this out?
Anyway this is what I did. Suppose that $a < 9b$. (Need to check what should be corrected if $a > 9b$).
Step 1: Comput... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
How to calculate the integral $\int\frac{1}{\sqrt{(x^2+8)^3}}dx$? I need to solve something like this
$$\int\frac{1}{\sqrt{(x^2+8)^3}}dx$$
Wolfram alpha says the solution is $$\frac{x}{8\sqrt{x^2+8}} + c$$
The problem is that the integrand is obtained by the quotient rule:
$$\bigg(\frac{g(x)}{h(x)}\bigg)'=\frac{g'(x)h(... | As mentioned in another answer, the solution to the given integral can be found by making a substitution with a trigonometric function, then integrating, calculating, substituting back and simplifying.
For a general integral of the form
$I =\displaystyle \int \frac{1}{\left(a x^n+b\right)^p} \, dx$ ,
the solution can b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
How do I find the singular solution of the differential equation $y' = \frac{y^2 + 1}{xy + y}$? I start out with the separable differential equation,
$$y' =\frac{dy}{dx} = \frac{y^2 + 1}{xy + y} = \frac{y^2 + 1}{y(x+1)}$$
Thus, $\frac{1 }{x+1}dx = \frac{y }{y^2 + 1}dy$.
Then integrating both sides of the equation, I g... | I believe that $x + 1 = \pm e^C\sqrt{y^2 + 1}$ is the correct general solution. You can rewrite it as: $x + 1 = C\sqrt{y^2 + 1}$, where $C = \pm e^c \neq 0$.
As for your concern about the singular solution, I don't think the original equation has $x = -1$ as the singular solution because $x = -1$ makes the original equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2655177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Maximum value of $5\sin x - 12 \cos x + 1$ Given that $5\sin x - 12\cos x = 13\sin (x-67.4)$
Find the maximum value of $5\sin x - 12 \cos x + 1 $ and the corresponding value of x from 0 to 360.
Maximum value = $13+1=14$
Corresponding value of $x$
$13\sin (x-67.4) + 1 = 14$
$\sin(x-67.4) = 1 $
$x = 157.4 , 337.4 $ ... | Note that we have the maximum value for $x-67.4°=90°$ indeed for $x-67.4°=270°$ we have that $\sin=-1$ is minimum.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to extract $(x+y+z)$ or $xyz$ from the determinant Prove
$$\color{blue}{
\Delta=\begin{vmatrix}
(y+z)^2&xy&zx\\
xy&(x+z)^2&yz\\
xz&yz&(x+y)^2
\end{vmatrix}=2xyz(x+y+z)^3}
$$
using elementary operations and the properties of the determinants without expanding.
My Attempt
$$
\Delta\stackrel{C_1\rightarrow C_1+C_2+C_3... | Let $s=x+y+z$, then
$$
\begin{align}
&\det\begin{bmatrix}
(y+z)^2&xy&zx\\
xy&(z+x)^2&yz\\
zx&yz&(x+y)^2
\end{bmatrix}\\[9pt]
&=\det\begin{bmatrix}
s(y+z)&xy&zx\\
s(z+x)&(z+x)^2&yz\\
s(x+y)&yz&(x+y)^2
\end{bmatrix}\tag1\\[9pt]
&=\det\begin{bmatrix}
2s^2&s(z+x)&s(x+y)\\
s(z+x)&(z+x)^2&yz\\
s(x+y)&yz&(x+y)^2
\end{bmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Probability of at least 3 identical numbers from an array I have an array {1, 2, 3, 4, 5} and I extract 4 random numbers. I want to find out the probability that at least 3 numbers are identical.
I concluded that it's the probability of having exactly 3 identical numbers added to the probability of having exactly 4 ide... | You calculated the probability of getting at least three of a $particular$ number. However, we can get at least three of $any$ of the five numbers.
Let $X$ denote the number of identical numbers. First we compute $P(X=3)$.
We must get three of one number and one of another. First choose one of the five numbers to be ob... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integral $\int-\sin x\mathrm e^{-\sin x}$ I want to integrate the function $$-\sin x\mathrm{e}^{-\sin x}.$$
I tried integration by parts with $u=-\sin x$ but I got another difficult integration which is $$\int (\cos x)^{2}\mathrm{e}^{-\sin x} \,\mathrm{d}x.$$
Could someone help me out?
| $-\int\mathrm e^{-\sin x}\sin x~dx=\int\sum\limits_{n=0}^\infty\dfrac{\sin^{2n+2}x}{(2n+1)!}dx-\int\sum\limits_{n=0}^\infty\dfrac{\sin^{2n+1}x}{(2n)!}dx$
For $n$ is any non-negative integer,
$\int\sin^{2n+2}x~dx=\dfrac{(2n+2)!x}{4^{n+1}((n+1)!)^2}-\sum\limits_{k=0}^n\dfrac{(2n+2)!(k!)^2\sin^{2k+1}x\cos x}{4^{n-k+1}((n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Verify the Limit: $\lim_{x\rightarrow 9} (\sqrt{x} +9) = 12$ I am using the definition of the limit of a function. So far I have:
$$|f(x)-L|<\epsilon \iff |(\sqrt{x}+9)-12|<\epsilon \iff |\sqrt{x}-3|<\epsilon.$$
This is where I get stuck. Any help is appreciated.
| Keep in mind that in deriving these proofs we are always working backwards. We start with $\epsilon$ and try to find $\delta$ that will make it true.
Let $\epsilon > 1$. Since $\epsilon $ may be as small as we like we may as well assume $1 > \epsilon > 0$. I'm saying this now, because later we will have a reason for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Calculating the summation $\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}$ I need to find explicitly the following summation
$$\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}, \quad
H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$$
From Mathematica, I checked that the answer is $2$. The same result is returned by WolframAlpha.
A thought ... | Noting that
$$\frac{1}{n(n + 1)} = \frac{1}{n} - \frac{1}{n + 1},$$
the sum may be rewritten as
$$\sum_{n = 1}^\infty \frac{H_{n+1}}{n(n + 1)} = \sum_{n = 1}^\infty \left (\frac{H_{n + 1}}{n} - \frac{H_{n + 1}}{n + 1} \right ).\tag1 \label1$$
Now as the harmonic numbers satisfy the recurrence relation
$$H_{n + 1} = H_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
real value of $k$ inirrational equation Find value of $k$ for which the equation $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}$ has no solution.
solution i try $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}......(1)$
$\displaystyle \sqrt{3z+3}+\sqrt{3z-9}=\frac{12}{\sqrt{2z+k}}........(2)$
$\displaystyle 2\sqrt{3z+3}=\frac{12}{\sqrt{2z+k... | Since $z\geq3$, we obtain:
$$\sqrt{3z+3}=\sqrt{2z+k}+\sqrt{3z-9}=\sqrt{5z+k-9+2\sqrt{(2z+k)(3z-9)}}=$$
$$=\sqrt{3z+3+k-6+2(z-3)+2\sqrt{(2z+k)(3z-9)}}\geq\sqrt{3z+3+k-6},$$
which gives that for $k>6$ our equation has no roots.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Creating 6 Digit Numbers Whose Adjacent Digits are Relatively Prime Six-digit integers will be written using each of the digits $1$ through $6$ exactly once per six-digit integer. How many different positive integers can be written such that all pairs of consecutive digits of each integer are relatively prime? (Note: $... | If $6$ is an interior entry then it has to appear in one of the subwords $165$ or $561$. If all three of $\{2,3,4\}$ are on the same side of these the digit $3$ has to be in the middle of the three (gives $4$ ways). If one of $\{2,3,4\}$ stands alone it has to be one of $2$ and $4$ (gives another $8$ ways). In all ther... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Proving that if $x>0$ then $\frac{x-(x^2+1)\arctan(x)}{x^2(x^2+1)}$ is less than $0$ How do I show that for $x>0$:
$$\frac{x-(x^2+1)\arctan(x)}{x^2(x^2+1)} < 0$$
I tried to do it somehow using the fact that $$\frac{\arctan(x)}{x} < 1$$ but still didn't figure it out...
| We have $$\begin{align}\frac{x-(x^2+1)\arctan x}{x^2(x^2+1)} < 0&\iff x<(x^2+1)\arctan x\\&\iff (1+x^2)\arctan x-x>0\end{align}$$ This is true iff $$\frac d{dx}\left[(1+x^2)\arctan x-x\right]=2x\arctan x+\frac{1+x^2}{1+x^2}-1=2x\arctan x>0$$ which is true since both $x$ and $\arctan x$ are greater than $0$ when $x>0$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Roots of $f(x)=x^3-x^2-2x+1$ We can prove using a monotony study that the function $f(x)=x^3-x^2-2x+1$ has three real roots. However, when I solve the equation $f(x)=0$ using Mathematica, I get
$$x_1=\frac{1}{3}+\frac{7^{2/3}}{3 \left(\frac{1}{2} \left(-1+3 i \sqrt{3}\right)\right)^{1/3}}+\frac{1}{3} \left(\frac{7}{2} ... | This is Example 9.2.2 in the first edition of Galois Theory by David A. Cox. He gives the polynomial, on page 239, as $y^3 + y^2 - 2 y -1.$
Easy enough to confirm the next bit with trig identities for $\cos 2t$ and $\cos 3t$ in terms of $\cos t \; .$ We have $\cos 2t = 2 \cos^2 t - 1$ and $\cos 3t = 4 \cos^3 t - 3 \cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Show that $\frac{1}{a^n+b^n+c^n} = \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}$ Let $n \in N, n=2k+1, and \text{ } \frac{1}{a+b+c} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$.
Show that $$\frac{1}{a^n+b^n+c^n} = \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}$$
I have tried, but I don't get anything. Can you please gi... | This method of infinite ascent is probably wrong.. but...
Suppose none of $a = -b, b = -c, c = -a$ is true.
Then multiplying both sides by $abc$ and inverting we get:
$\frac{a+b+c}{abc} = \frac{1}{ab + bc + ca} \implies \frac{1}{ab + bc + ca}= \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}$
We can now write $x = ab, y = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2668597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Calculating a multivariable limit. Find if it exists the limit : $$\lim_{(x,y)\to(0,0)}\frac{x^4y^4}{(x^2+y^4)^3}$$
I've tried the following :
1st attempt : Using polar coordinates:
Set $x= r\cos\theta$ and $y=r\sin\theta$
$$\frac{x^4y^4}{(x^2+y^4)^3}=\frac{r^8\cos^4\theta \sin^4\theta}{(r^2\cos^2\theta+r^4\sin^4\theta... | Approaching along the path $x = y^2$ (motivated by trying to simplify $x^2 + y^4$, we have
$$\lim_{y \to 0} \frac{(y^2)^4 y^4}{((y^2)^2 + y^4)^3} = \lim_{y \to 0} \frac{y^{12}}{(2y^4)^3} = \frac 1 8 \ne 0.$$
Hence, the limit does not exist.
Alternative interpretation in polar coordinates: $\cos \theta \to 0$ in such a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Angle between diagonals of an irregular pentagon
In pentagon $ABCDE$, $DC$ is parallel to $BE$ and $BC$ is parallel to $AE$. $△DCE$ and $△BCE$ are both isoscles with $$∠CBE= ∠DEC = α,\ ∠EAB= α+\frac{π}{2}.$$ Suppose $AD$ meets $BE$ at $G$. Find $∠BGA$.
I have tried producing $AD$ to meet $BC$ at $F$ and showing that... | Let $DC=ED=a$.
Hence, $$EC=CB=2a\cos\alpha$$ and
$$EB=2EC\cos\alpha=4a\cos^2\alpha.$$
Thus, by law of sines for $\Delta EAB$ we obtain:
$$\frac{EA}{\sin\left(90^{\circ}-2\alpha\right)}=\frac{4a\cos^2\alpha}{\sin\left(90^{\circ}+\alpha\right)},$$
which gives $$EA=4a\cos\alpha\cos2\alpha.$$
Now, since $\alpha<60^{\circ}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Surd inside Surd equality I was trying a problem where I got the following surd as my answer:
$$ {\sqrt{6 - 2 \sqrt{5} }\over 4} \approx 0.309016.... $$
The answer listed was:
$$ {\sqrt{5} - 1 \over 4} \approx 0.309016.... $$
Is there a simple way that you could get to the second expression from the first?
| Suppose you assume that you can write $\sqrt{6-2\sqrt5} = \sqrt a - \sqrt b$ for some rational numbers $a$ and $b$. Squaring both sides will give you $6-2\sqrt5 = a+b-2\sqrt{ab}$, and so
\begin{align*}
6 &= a+b \\
\sqrt{5} &= \sqrt{ab}
\end{align*}
So we want $ab=5$ and $a+b=6$. An obvious solution is, indeed, $a=5$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Greatest Common Divisor of two Polynomials. Find the $\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$ over $\mathbb{Q}[x]$. Then find two polynomials $a(x),b(x) \in \mathbb{Q}[x]$ such that, $$a(x)(x^3-6x^2+14x-15) + b(x)(x^3-8x^2+21x-18)=\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$$
I have managed to find,
$$x^3-6x^2+14x-15=(x-... | You have the complete extended Euclid algorithm (which IMHO is the best way) given already in Will Jaggy's post.
Given that you already extracted $x-3$ as GCD, here is a perhaps quicker way (which may not generalise much). We simplify a bit by noting we seek linear $a(x), b(x)$, s.t.:
$a(x)\color{blue}{(x^2-3x+5)} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Finding $\lim_{x\to \infty}(1+\frac{2}{x}+\frac{3}{x^2})^{7x}$ I am having some diffuculty finding the limit for this expression and would appreciate if anyone could give a hint, as to how to continue. I know the limit must be $e^{14}$ (trough an engine) and I can show it for
$(1+\frac{2}{x}+\frac{1}{x^2})^{7x}$ like ... | It can be convenient to use Taylor series when doing problems like this:
$$\lim_{x \to \infty} \left(1 + \frac{2}{x} + \frac{3}{x^2}\right)^{7x} = \lim_{x \to \infty} e^{7x\ln(1+\frac{2}{x}+\frac{3}{x^2})} = \lim_{x\to \infty} e^{7x\left( \frac{2}{x} + \frac{3}{x^2} + \mathcal{O}\left(\left((\frac{2}{x} + \frac{3}{x^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
If $\sum\limits_{k=1}^n\frac1{x_k+y_k}\leqslant\frac n2$, prove that $\sum\limits_{k=1}^n(\frac{x_k}{1+y_k}+\frac{y_k}{1+x_k})\geqslant n$
Let $n \in \mathbb{N}, n\geqslant 2$ and $x_1,x_2,\cdots,x_n,y_1,y_2,\cdots,y_n>0$ with $$\sum_{k=1}^{n}{ 1 \over{x_k+y_k} } \leqslant {n\over 2}.$$
Show that $$\sum_{k=1}^{n}\l... | For each $k$,\begin{align*}
\frac{x_k}{1 + y_k} + \frac{y_k}{1 + x_k} &= \frac{(x_k + y_k + 1)(x_k + y_k + 2)}{(1 + x_k)(1 + y_k)} - 2\\
&\geqslant \frac{(x_k + y_k + 1)(x_k + y_k + 2)}{\dfrac{1}{4}((1 + x_k) + (1 + y_k))^2} - 2 = \frac{2(x_k + y_k)}{x_k + y_k + 2},
\end{align*}
thus$$
\sum_{k = 1}^n \left( \frac{x_k}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What are the maximum and minimum values of $4x + y^2$ subject to $2x^2 + y^2 = 4$? $$ 2x^2 + y^2 = 4 $$
$$ Y = \sqrt{4-2x^2} $$
$$4x + y = 2x^2 + \sqrt{4-2x^2}$$
Find the derivative of $$ 2x^2 + \sqrt{4-2x^2} $$ set as = 0
$$X^2 = 64/33$$
$$ F(64/33) = 34\sqrt{33}/33 $$
How to solve it the right way?
| you will get $$4x+y^2=4x+4-2x^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2687820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
A first course in Complex Analysis with Applications Assume for the moment that $\sqrt{1+i}$. makes sense in the complex number system.
How would you then demonstrate the validity of the equality
$$\sqrt{1+i} = \sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{2}} + i\sqrt{-\frac{1}{2} + \frac{1}{2} \sqrt{2}}$$?
Source: Dennis G.... | In order to prove o$$\sqrt{1+i} = \sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{2}} + i\sqrt{-\frac{1}{2} + \frac{1}{2} \sqrt{2}}$$
we square the right hand side and show that it is the same as $1+i.$
Note that $$\bigg (\sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{2}} + i\sqrt{-\frac{1}{2} + \frac{1}{2} \sqrt{2}}\bigg )^2 =$$
$$ \bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2690046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simplifying $\frac{y}{y^2+b^2}$ where $y=b \cot\theta$ Could appreciate some help with this question.
I want to simplify the following trigonometric equation.
$$\frac{y}{y^2+b^2}$$
where $y=b \cot\theta$.
The solution I got was
$$\frac{1}b \cos\theta$$
Can someone verify and try and guide me through the solution... | Hint:
$$b^2+b^2\cot^2(x)=b^2(1+\cot^2(x))=b^2\left(1+ \frac{\cos^2(x)}{\sin^2(x)}\right)=b^2\left(\frac{\sin^2(x)}{\sin^2(x)}+ \frac{\cos^2(x)}{\sin^2(x)}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2690581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
$6$ digit numbers formed from the first six positive integers such that they are divisible by both $4$ and $3$. The digits $1$, $2$, $3$, $4$, $5$ and $6$ are written down in some order to form a six digit number. Then (a) how many such six digits number are even? and (b) how many such six digits number are divisible b... | Since digits are not repeated, all $6$ digits must be used.
Sum of all digits is $21$ $(1+2+3+4+5+6)$. So this number is divisible by $3$.
Now we only have to check its divisibility by $4$.
Number should end with any of $8$ combination $(12,16,24,32,36,52,56,64)$.
Starting $4$ digits can be in any order.
So total count... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2690698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Calculate $ \arctan(x_{1}) \cdot \arctan(x_{2}) $
Let $ x_{1} $ and $ x_{2} $ be the roots of the equation : $ x^2-2\sqrt{2}x+1=0 $
Calculate $ \arctan(x_{1}) \cdot \arctan(x_{2}) $
The answer should be $ \dfrac{3\pi^2}{64} $.
How does the fact that $ x_{1} $ = $ 1 + \sqrt2 $ = $ \dfrac{1}{x_{2}} $ help?
| We have $$x^2-2\sqrt{2}x+1=(x-(1+\sqrt2))(x-(-1+\sqrt2))=0$$ so let $x_1=1+\sqrt2$ and $x_2=-1+\sqrt2$.
Let $a=\tan^{-1}(1+\sqrt2)$. Using the double angle tangent formula, $$\begin{align}\tan2a=\frac{2\tan a}{1-\tan^2a}&\implies\tan2a=\frac{2(1+\sqrt2)}{1-(1+\sqrt2)^2}=-1\end{align}$$ and hence taking the principal an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2692518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Is $x^2 \equiv 1 \pmod{p^k} \iff x \equiv \pm 1 \pmod {p^k}$ for odd prime $p$? Problem: As the title says, is $x^2 \equiv 1 \pmod {p^k} \iff x \equiv \pm 1 \pmod {p^k}$ for odd prime $p$? If not, is it at least true that there are two solutions to the congruence equation?
Attempt: I am a beginner in number theory, and... | Assume $x^2 \equiv 1 \pmod {p^k}$. Then $x^2-1 \equiv 0 \pmod {p^k}$, meaning $p^k | x^2-1=(x-1)(x+1)$. Since $p$ is prime, we can conclude that $p^k\vert(x-1)$ or $p^k\vert(x+1)$. Yet, $p^k\vert(x-1) \Rightarrow x-1 \equiv 0 \pmod {p^k} \Rightarrow x \equiv +1 \pmod {p^k}$ and $p^k\vert(x+1) \Rightarrow x+1 \equiv 0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
$a+b+c=?$ by having $a^2+160=b^2+5$ and $a^2+320=c^2+5$ My question:
$a+b+c=?$ by having $a^2+160=b^2+5$ and $a^2+320=c^2+5$.
My work so far:
$a^2+160=b^2+5\Rightarrow (b-a)(a+b)=155=31\times 5$
$a^2+320=c^2+5\Rightarrow (c-a)(c+a)=315=5\times3^2\times 7$
And now, I'm stuck.
($a,b,c$ are a members of $\mathbb Z$ and ar... | Hint:
So, $a+b> a-b$ and $a+b=\dfrac{31\cdot5}{a-b}$
Either $a+b=155, a-b=?$ or $a+b=31,a-b=?$
Check which values of $a$ keep $c$ integer
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Solve $\frac{x^2}{\left(a+\sqrt{a^2+x^2}\right)^2}+\frac{x^2(1+x^2)}{\left(a+\sqrt{a^2+x^2(1+x^2)} \right)^2}=1$ I have been trying to solve the following equation for $x$:
\begin{align}
\frac{x^2}{\left(a+\sqrt{a^2+x^2}\right)^2}+\frac{x^2(1+x^2)}{\left(a+\sqrt{a^2+x^2(1+x^2)} \right)^2}=1,
\end{align}
for some fixed ... | Hints:
Let $x = a \tan (2 A)$ and $x\sqrt{1 +x^2} = a \tan (2 B)$, then, due to the $\tan$- half-angle theorem, the equation in question transforms into
$$
\tan^2 (A) + \tan^2(B) = 1
$$
This can be used for numerical analysis, e.g. (with fixed $a$) a Banach type fixed point iteration $x^{(n)} \to B \to A \to x^{(n+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Are there any "nonstandard" special angles for which trig functions yield radical expressions? Everyone learns about the two "special" right triangles at some point in their math education—the $45-45-90$ and $30-60-90$ triangles—for which we can calculate exact trig function outputs. But are there others?
To be specifi... | Yes, a $15-75-90$ triangle may be the one you want.
Assume we have a right $\Delta ABC$ with $\widehat{BAC}=15^0;\widehat{ABC}=90^0;\widehat{ACB}=75^0$.
Put an extra point $D$ like above so that $B,C,D$ are collinear and $AC$ is the angle bisector of $\widehat{DAB}$, this means $\widehat{DAB}=30^0;\widehat{BDA}=60^0$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 9,
"answer_id": 5
} |
$\left|m^2 -\frac{n\cdot(n+1)}{2}\right|=1$ and A001110 The pairs of squares and triangular numbers differing by $1$ are so far: $(0,1)$, $(3,4)$, $(9,10)$, $(15,16)$, $(120,121)$, $(528,529)$. Do you think there are more terms than are found in A001110? If so, could there be a reason for this?
| Let's start with an existing solutuion $(m,n)$:
$$\left|m^2 -\frac{n\cdot(n+1)}{2}\right|=1 \tag{1}$$
Then
$$\left|(m+1)^2 -\frac{n\cdot(n+1)}{2}-(2m+1)\right|=1$$
$$\left|(m+2)^2 -\frac{n\cdot(n+1)}{2}-(2m+1)-(2m+3)\right|=1$$
$$\left|(m+3)^2 -\frac{n\cdot(n+1)}{2}-(2m+1)-(2m+3)-(2m+5)\right|=1$$
$$... \text{by induct... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2696312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the rational solution of the equation Find the rational solution of the equation:
$\frac{2x - 1}{2016} + \frac{2x - 3}{2014} + \frac{2x - 5}{2012}+ ...+ \frac{2x - 2011}{6} +\frac{2x - 2013}{4} + \frac{2x - 2015}{2} =\\ \frac{2x - 2016}{1} + \frac{2x - 2014}{3} + \frac{2x - 2012}{5}+ ...+ \frac{2x - 6}{2011} + \fr... | Alpha gives $x=\frac {2017}2$. With that value all the fractions become $1$ so each side sums to $1008$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2698770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Show that $\frac{1}{3}<\int_0^1\frac{1}{1+x+x^2} \, dx <\frac{\pi}{4}$ Show that $$\frac{1}{3}<\int_0^1\frac{1}{1+x+x^2}\,dx <\frac{\pi}{4}$$
I want to use if $f<g<h$ then $\int f<\int g<\int h$ formula for Riemann integration.
$1+x^2<1+x+x^2$ and it will give RHS as $$\frac{1}{1+x+x^2}<\frac{1}{1+x^2}$$
How to choose... | A better upper bound: by the convexity of the integrand in $[0,1]$:
$$\forall x\in[0,1]:\qquad\frac1{1 + x + x^2}\le 1 - \frac{2x}3,$$
and this implies
$$\int_0^1\frac{1}{1+x+x^2}dx\le\int_0^1(1 - 2x/3)\,dx = \frac23.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2699005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
What kind of matrix is this and why does this happen? So I was studying Markov chains and I came across this matrix \begin{align*}P=\left( \begin{array}{ccccc}
0 & \frac{1}{4} & \frac{3}{4} & 0 & 0\\
\frac{1}{4} & 0 & 0 & \frac{1}{4} & \frac{1}{2}\\
\frac{1}{2} & 0 & 0 & \frac{1}{4}& \frac{1}{4... | Notice that your matrix $$\begin{align*}P=\left( \begin{array}{ccccc}
0 & \frac{1}{4} & \frac{3}{4} & 0 & 0\\
\frac{1}{4} & 0 & 0 & \frac{1}{4} & \frac{1}{2}\\
\frac{1}{2} & 0 & 0 & \frac{1}{4}& \frac{1}{4}\\
0 & \frac{1}{4} & \frac{3}{4} & 0& 0\\
0 & \frac{1}{4} & \frac{3}{4} & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2699410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 5,
"answer_id": 3
} |
Prove that the area of $\triangle DEF$ is twice the area of $\triangle ABC$ Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point on its circumcircle. Let lines $PA$ and $BC$ intersect at $D$ , let lines $PB$ and $CA$ intersect at $E$, and let lines $PC$ and $AB$ intersect at $F$. Prove that the area o... | Proof (for the case when $P$ lies on arc $BC$):
Let $AB=BC=CA=1$ and $\angle PAC=\theta$.
$$\frac{PD}{\sin\angle PBD}=\frac{PB}{\sin \angle PDB} \implies \frac{PD}{\sin\theta}=\frac{PB}{\sin (120^\circ-\theta)} $$
$$\frac{PB}{\sin\angle PAB}=\frac{AB}{\sin \angle APB}\implies \frac{PB}{\sin(60^\circ-\theta)}=\frac{AB}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2701381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Prob. 10, Sec. 3.3, in Bartle & Sherbert's INTRO TO REAL ANALYSIS, 4th ed: Is this sequence convergent? Here is Prob. 10, Sec. 3.3, in the book Introduction to Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition:
Establish the convergence or the divergence of the sequence $\left( y_n \right)$, where
... | Your proof is correct.
It is too early (in the book) to obtain the limit. Some time later it will be easy to see an integral sum of
$$
\int_{0}^{1}\frac{1}{1+x}\,dx.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
How do I calculate this kind of expressions $(-1)^{-\frac{2}{5}}$? I used a tool and found that it is equal to $0.31 - 0.95i$
Can someone please tell me the way I reach this result step by step?
| That's only one possible value.
$(-1)^{-\frac 25}$ will be a number $z$ so that $z^5 = (-1)^{-2} = 1$. $z = 1$ is one such number. $z = 1$ is the only real such number. There are four other such complex numbers.
If we view a non-zero complex number, $a + bi$ as a point $(a,b)$ in a plane the point has a distinct dis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2706275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
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.