Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Finding the limit of a function with ArcTan I've found difficulties finding this limit ( without using Taylor series approximation, as it's intended for the secondary-school ):
$$
\lim_{x\ \to\ \infty}\left[\,
{x^{3} \over \left(\,x^{2} + 1\,\right)\arctan\left(\,x\,\right)} - {2x \over \pi}
\,\right]
$$
Thanks.
| Hint. You may write, as $x$ tends to $+\infty$,
$$
\arctan x=\frac{\pi}{2}-\arctan \frac{1}{x}
$$
and use $$ \arctan u=u-\frac{u^3}{3}+\mathcal{O}(u^4), \quad u \rightarrow 0,$$
to obtain $$
(x^2+1)\arctan x=(x^2+1)\left(\frac{\pi}{2}-\arctan \frac{1}{x}\right)=\frac{\pi x^2}{2}-x+\frac{\pi }{2}+\mathcal{O}\left(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1035815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Prove determinant is zero If
$M = \begin{vmatrix}
1 & a & b+c \\
1 & b & a+c \\
1 & c & a+b \\
\end{vmatrix}$
Show that M = 0 WITHOUT expanding the determinant.
I have tried row operations and haven't had much success. Any tips?
| $$\begin{vmatrix}1&a&b+c\\
1&b&a+c\\
1&c&a+b\end{vmatrix}\stackrel{R_2-R_1\,,\,\,R_3-R_1}\longrightarrow\begin{vmatrix}1&a&b+c\\
0&b-a&a-b\\
0&c-a&a-c\end{vmatrix}= $$
$${}$$
$$=(a-b)(a-c)\begin{vmatrix}1&a&b+c\\
0&-1&1\\
0&-1&1\end{vmatrix}$$
and clearly we reached two equal rows
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1036202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Theorem for Equal Sums of Like Powers $x_1^8+x_2^8+x_3^8+\dots$ Kindly see the question at the end of post. Solutions to the system of three equations,
$$\begin{aligned}
a^2+b^2+c^2+d^2\, &= e^2+f^2+g^2+h^2\\
a^4+b^4+c^4+d^4\, &= e^4+f^4+g^4+h^4\\
abcd\, &= efgh
\end{aligned}\tag{1}$$
satisfy two nice consequences.
Co... | What I write here is not the final answer to this question, but I think it will be helpful.
I notice that the Consequence 1 can be simplified and generalized as below,
Denote
$$R_n=(a^n+b^n+c^n+d^n-e^n-f^n-g^n-h^n)/n$$ for any $n<>0$, and
$$R_0=2(abcd-efgh)/(abcd+efgh)$$
$$m=(a^2+b^2+c^2+d^2)/(a+b+c+d)^2$$
We have
If... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1037126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is the determinant of this matrix positive or negative? $\left( \begin{array}{ccc}
1 & 1000 & 2 & 3 &4\\
5 & 6 &7&1000 &8\\
1000&9&8&7&6\\
5 & 4&3&2&1000\\
1&2&1000&3&4\\ \end{array} \right)$
When I compute the determinant online, I find that it is positive, but I'm supposed to "see" something about the matrix that all... | By definition, the determinant is the sum of $60$ positive and $60$ negative terms, one of which is $\mbox{sgn}{\left(
\begin{array}{ccccc}
1 & 2 & 3 & 4 & 5 \\
3 & 1 & 5 & 2 & 4 \\
\end{array}
\right)}(1000)^5=\mbox{sgn}\left((13542)\right)(1000)^5=+(1000)^5$.
Each of the other terms has absolute value at most $9\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1039023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
I have proven that $e^x > x^3$ for $x>5$, can I prove that $\lim \frac{x^3}{e^x} = 0$? In order to calculate the limit
$$\lim_{x\to\infty} \frac{x^3}{e^x} = 0$$
I've verified that:
$$f(x) = e^x-x^3\\f'(x) = e^x-3x^2\\f''(x) = e^x-6x\\f'''(x) = e^x-6$$
Note that $x>3 \implies f'''(x)>0$, therefore $f''(x)$ is crescent. ... | $$\lim_{x \to 0} \frac{x^3}{e^x} = \lim_{x \to 0} \frac{x^3}{1 + x + x^2/2! + x^3/3! + \dots} = \frac 01 = 0$$
And
$$\begin{align}\lim_{x \to \infty} \frac{x^3}{e^x}
&= \lim_{x \to \infty} \frac{x^3}{1 + x + x^2/2! + x^3/3! + x^4/4!+\dots}
\\&= \lim_{x \to \infty} \frac{1}{x^{-3} + x^{-2} + x^{-1}/2! + 1/3! + x/4!+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1045132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Application of Mergesort We have $8$ players and we want to sort them in $24$ hours.
There is one stadium. Each game lasts one hour.
In how many hours can we sort them??
I thought that we could it as followed:
$$\boxed{P1} \ \boxed{P2} \ \boxed{P3} \ \boxed{P4} \ \boxed{P5} \ \boxed{P6} \ \boxed{P7} \ \boxed{P8} \\ ... | I think that the question is: if we wish to sort $n$ items by comparing them two at a time, and the algorithm is completely determined by the results of those comparisons, what is the minimum number of comparisons that we need to make?
For example, if there are $n = 3$ items, there are 6 possible arrangements, and we n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1045411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Maximum and minimum of $f(x,y)=xy$ when $x^2 + y^2 + xy =1$ It is asked to find the maximum and minimum points of the function
$$f(x,y)=xy$$
when $x^2 + y^2 + xy=1$
I've tried Lagrange and obtained
$$\lambda = \frac{y}{2x+y}=\frac{x}{2y+x}$$
but what should I do with this? Any other suggestion?
Thanks!
| you can do this without calculus. the $x$-coordinates of the points common to both $xy = k$ and $x^2 + y^2 + xy = 1$ satsfies $ x^2 + k^2/x^2 + k - 1 = 0$ which can be turned into a quadratic equation for $u = x^2$ as $$u^2 + (k-1)u^2 + k^2 = 0$$ whose discriminant $ 1 - 2k - 3k^2 $ is positive for $-1 \le k \le 1/3.$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1045490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Check: Find all the Sylow $3$-subgroups of $A_4$. Find all the Sylow $3$-subgroups of $A_4$.
The order of $A_4$ is $\frac{4!}{2}=\frac{24}{2}=12=3\cdot 2^2$. So the Sylow $3$-subgroups are
*
*$\{ \begin{pmatrix} 1 \end{pmatrix}, \begin{pmatrix} 2 & 3 & 4 \end{pmatrix}, \begin{pmatrix} 2 & 4 & 3 \end{pmatrix} \}$
*$... | The highest power of 3 that divides $|A_4|$ is $3^1=3$. So we need to list all subgroups of $A_4$ of order 3. One of them is clearly $\langle (234) \rangle$, as you mention. Now we need to determine all the remaining subgroups of order 3. One way is as follows. Since a group of order 3 is cyclic, a subgroup of orde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1045622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to simplify this expression? $\frac{x+1}{x^2+x-2}-\frac x{x^2-1}$. $$\frac{x+1}{x^2+x-2}-\dfrac x{x^2-1}=\text{ ?}$$
This expression needs to be simplified. I've tried to do it but i couldn't.
| Hint;
$x^2+x-2=(x-1)(x+2)\tag{1}$
$x^2-1=x^2-1^2=(x-1)(x+1)\tag{2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1046985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove by induction that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{n+2}{2^n}$ Prove by induction that
$$
\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{n+2}{2^n}
$$
Let $n=r$, so that
$$
S_r=2-\frac{r+2}{2^r}
$$
Therefore
$$\begin{align}
S_{r+1}=S_r+\frac{r+1}{2^{r+1}}&... | Note that
$$-\frac{r+2}{2^r}+\frac{r+1}{2^{r+1}}=\frac{-2(r+2)}{2^{r+1}}+\frac{r+1}{2^{r+1}}=-\frac{r+3}{2^{r+1}}=-\frac{(r+1)+2}{2^{r+1}}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1047277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$x^n + y^n = c$ has finitely many integral solutions? Assume $n > 1$ and $n$ is odd because it's easy if $n$ is even.
Please help prove this.
$x^n + y^n = c$ has finitely many integral solutions if $c \neq 0$?
Thank you all for replying. I think I've just found my answer to this one. And I think it's quite simple. Her... | Let's look at the factorization of $x^n+y^n$.
For all odd $n$, $$x^n+y^n=(x+y)\left(x^{n-1}-x^{n-2}y+x^{n-3}y^2-x^{n-4}y^3+\cdots-xy^{n-2}+y^{n-1}\right)$$
Thus, since $x^n+y^n=c$, $$\left(x^{n-1}-x^{n-2}y+\cdots-xy^{n-2}+y^{n-1} \right)\Bigg| c$$
This implies that $$\left|x^{n-1}-x^{n-2}y+\cdots-xy^{n-2}+y^{n-1}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Asymptotic development of a recurrent sequence Let $u_0 = 1$ and $u_{n+1} = \frac{u_n}{1+u_n^2}$ for all $n \in \mathbb{N}$.
I can show that $u_n \sim \frac{1}{\sqrt{2n}}$, but I would like one more term in the asymptotic development, something like $u_n = \frac{1}{\sqrt{2n}}+\frac{\alpha}{n\sqrt{n}} + o\bigl(\frac{1}{... | Let $a_n=\dfrac1{u_n^2}$. Then
$$
a_{n+1}-a_n = \left(\frac{1+u_n^2}{u_n}\right)^2 - \frac1{u_n^2} = 2+u_n^2 = 2+\frac1{a_n}.
$$
From this and $a_2=4$ we can find a successive sequence of estimates for $n\ge2$:
\begin{align*}
a_n &\ge 2n; \\
a_n &\le 2n + \sum_{k=2}^{n-1}\frac1{2k}
< 2n+\frac12\log n; \\
a_n &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Differentiation with the quotient rule I have the question:
Given that $$y=\frac{e^x}{\sqrt{1+2x}}$$
Show that $$\frac{dy}{dx} = \frac{2xe^x}{\sqrt{(1+2x)^3}}$$
I've done the question but I got $2xe^x(\sqrt{(1+2x)^3})$. I feel like this is too similar to the sheet's answer, so am I wrong or is the sheet printed wrong?
... | I suspect
$$\frac{1}{\left(\sqrt{1+2x}\right)^2} \not = \frac{1}{\left({1+2x}\right)^2}$$
and $$\frac{\frac{1}{\sqrt{1+2x}}}{(1+2x)^2} \not =\frac{1}{(1+2x)^{-\frac{3}{2}}}$$
but $$\frac{\frac{1}{\sqrt{1+2x}}}{\left(\sqrt{1+2x}\right)^2} = \frac{1}{\left(\sqrt{1+2x}\right)^3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1050478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Help with two limit problems I have to prove these limits by definition. Can somebody help?
$\lim_{n \to \infty}\left ( \frac{n^2+1}{4n^2+5} \right )=\frac{1}{4}$
$\lim_{n \to \infty}\left ( \frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^n} \right )=1$
This is what i have done so far:
For the first I got this and i don't kno... | For the first one:
$$\left | \frac{n^2+1}{4n^2+5} -\frac{1}{4}\right |=\left | \frac{4n^2+4-4n^2-5}{4(4n^2+5)}\right |= \frac{1}{4(4n^2+5)}<\epsilon \\\iff 4n^2+5>\frac{1}{4\epsilon}\iff 4n^2>\frac{1}{4\epsilon}-5\iff n^2>\frac{1}{16\epsilon}-\frac{5}{4}=\frac{1-5\epsilon}{16\epsilon} \\\iff n>\sqrt{\frac{1-5\epsilon}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1050595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Triple Integral to Find Volume Question: Use a triple integral to find the volume of the solid enclosed by the parabaloids $y=x^2+z^2$ and $y=8-x^2-z^2$.
My attempt: The best I can figure, this object looks kind of like a football oriented along the $y$-axis from $y=0$ to $y=8$ and is symmetric about the $y$-axis and t... | HINT:For a start, it is convenient to put $ x^2+z^2 = r^2$, so that you have solids of revolution:
$$ y_1=r^2 ,\ y_2=8-r^2 $$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1050979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Gemetric representation of complex numbers Find the geometric representation of;
|z-2| - |z+2| < 2
things i know;
| z - 2 | is the distance between a point z and the point (2,0) in the complex plane. It suppose to represent a hyperbola, BUT i have no idea on how to get there. i know the standard form of a hyperbola, bu... | If you want to manipulate it algebraically (without considering the geometric definition of the hyperbola), then let $$z=x+iy$$ and note that $$|z|^2=z\overline{z}=x^2+y^2.$$
Edit
First, as someone mentioned in the comments, I am not sure if you want the equation of the hyperbola or the set of points that satisfy your ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1053146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Combinatorics: premutations with repetition? I have following problem from combinatorics:
Let's have set of 8 distinct items: {a,b,c,d,e,f,g,h}
How many ways we can combine 10 of them if we know:
*
*We start with A and end with H
*Every item of the set must be in the result at least once.
I started as ... | Since $a$ must appear in the first of the ten positions, $h$ must appear in the last position, and each of the eight letters in the set $\{a, b, c, d, e, f, g, h\}$ must appear at least once, either one letter appears three times or two letters appear twice each.
Consider cases:
*
*The letter $a$ appears three tim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1055536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Possible values of $\lim_{x \rightarrow \infty} \left(\frac{p(x)}{q(x)}\right)^x$ I was asked by some freshmen the follwing two questions regarding the limit:
Using the fact that
$$\lim_{x \rightarrow \infty} \left(1 + \frac{1}{x}\right)^x = e$$
evaluate
$$\lim_{x \rightarrow \infty} \left(1 + \frac{3}{x^2}\right)^x$... | Let $c=a_{n-1}-b_{n-1}$. Assume $c\ne 0$.
Let $p(x)=(a_{n-1}-b_{n-1})^{-1}\left((a_{n-1}-b_{n-1}) x^{n-1} + \dots + a_0-b_0\right)$.
Let $q(x)=x^n + b_{n-1} x^{n-1} + \dots + b_0$.
Let $f(x)=\left(1+c\frac{p(x)}{q(x)}\right)^\frac{q(x)}{p(x)}$.
Let $g(x)=\frac{xp(x)-q(x)}{q(x)}$.
$$\left(\frac{x^n + a_{n-1} x^{n-1} + \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1056863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Given a matrix, find a matrix that satisfies Let A be a matrix (3x4)
Prove that there does not exists a matrix X that satisfies
$$
\begin{pmatrix}
1 & 1 & 2 & -1 \\
0 & 2 & 1 & 3 \\
1 & 1 & 2 & -1 \\
\end{pmatrix}X = \begin{pmatrix}
1 & 1 & 1 \\
0 & 2 & 0 \\
... | The rows of the LHS will be given by the rows of $A$, multiplied by $X$. Since the first and third rows of $A$ are the same, the first and third rows of the product will be the same. Therefore the product cannot equal the RHS.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1057337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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Find $\iint\limits_\Sigma(x\sqrt{x^2+y^2+z^2}\hat{\imath}+y\sqrt{x^2+y^2+z^2}\hat{\jmath}+z\sqrt{x^2+y^2+z^2}\hat{k})\cdot\hat{n}dS$
Find $\iint\limits_\Sigma(x\sqrt{x^2+y^2+z^2}\hat{\imath}+y\sqrt{x^2+y^2+z^2}\hat{\jmath}+z\sqrt{x^2+y^2+z^2}\hat{k})\cdot\hat{n}dS$ Where $\Sigma$ is the torus $\rho=\sin\phi$ Oriented ... | The equation for the torus is clearly directing you to switch to spherical coordinates $(\rho, \phi, \theta)$. The bounds are then $0 \le \rho \le \sin \phi$, $0 \le \phi \le \pi$, $0 \le \theta \le 2\pi$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1058752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find the first two non vanishing maclaurin terms Find the first two nonvanishing terms in the Maclaurin series of $\sin(x + x^3)$.
Suggestion: use the Maclaurin series of $\sin(y)$ and write $y = x + x^3$
Using
this result, find
$\lim\limits_{x\to 0}\frac{\sin(x + x^3)−x}{x^3}$
$\sin(y)= y-\frac{y^3}{3!}+\frac{y^5}{5!... | Note that the formula for Maclaurin Series is
$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n.$$
So calculating the first few terms with $y=x+x^3$ gives us
$$\sin(y)=\frac{0}{0!}y^0+\frac{1}{1!}y^1-\frac{0}{2!}y^2-\frac{1}{3!}y^3+\frac{0}{4!}y^4+...\\=y-\frac{y^3}{3!}+\frac{y^5}{5!}-...$$
Now, if we plug $x+x^3$ bac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1059711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Proving $1+2^n+3^n+4^n$ is divisible by $10$ How can I prove
$$1+2^n+3^n+4^n$$ is divisible by $10$ if $$n\neq 0,4,8,12,16.....$$
| We consider it $(\mathrm{mod}\;2)$ and $(\mathrm{mod}\;5)$ separately.
Clearly $1$ and $3^n$ are odd, while $2^n$ and $4^n$ are even, so their sum is even.
Now by Fermat's Little Theorem, when $a$ is not divisible by $5$, $a^5 \equiv a \mod 5$, so $a^4 \equiv 1 \mod 5$. Thus we only need to check $n=1$, $2$, and $3$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1060746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
Compare inequalities in a proof by induction I am solving a proof by induction example. But I ended up with my hypothesis
$$
a_{n-1} \geq \frac{2^n}{2}+n^2-2n+1
$$
and my inductive step
$$
a_{n-1} \geq \frac{2^n}{2}+\frac{n^2}{2}-\frac{n}{2}.
$$
How can I show that if I am assuming that my hypothesis is true then my in... | Alternatively, if you prefer a more direct approach, the finite difference method is really useful.
We have $$
\begin{array}{c|l|c|r}
n & a_n & b_n=a_{n+1}-a_{n} & c_n=(a_{n+2}-a_{n+1})-(a_{n+1}-a_{n}) \\
\hline
0 & 1 & 2 & 3\ \\
1 & 3 & 5 & 6\ \\
2 & 8 & 11& 12\ \\
3 & 19 & 23 & 24\ \\
4 & 42 & 47 & \vdots \\
5 & 89 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1063506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$ I used the following way but got wrong answer
$$A.M. \ge G.M.$$
$$ \frac{\sin \theta + \csc \theta}{2} \ge \sqrt{\sin \theta \cdot \csc \theta}$$
Squaring both sides,
\begin{equation*}
(\sin\theta + \csc\theta )^2 \ge 4 \tag{1}
\end{equation*}... | $$\sin^2\theta+\cos^2\theta+2+2+\csc^2\theta+\sec^2\theta$$
$$=5+\frac1{\sin^2\theta\cos^2\theta}=5+\frac4{(\sin2\theta)^2}=5+4\csc^22\theta$$
Now, $\csc^22\theta=1+\cot^22\theta\ge1$ for real $\theta$
The equality occurs if $\csc^22\theta=1\iff\sin^22\theta=1$
$\iff\cos2\theta=0\implies2\theta=(2n+1)\dfrac\pi2$ where ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1065943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Solve $x^7-5x^4-x^3+4x+1=0$ for $x$
Solve for $x$
$$x^7-5x^4-x^3+4x+1=0$$
This equation has been bugging me since the past few days. I have found, using the Rational Root Theorem that $x=1$ is a root of this equation. However, after dividing, I cannot solve the six degree equation thus generated. I have also trie... | Consider the identity
$(x^4-4x-1)^2=x^8-8x^5-2x^4+16x^2+8x+1$
Differentiating both sides w.r.t. $x$, we get,
$x^7-5x^4-x^3+4x+1=(x^4-4x-1)(x^3-1)$
Now, the equation becomes,
$(x^4-4x-1)(x^3-1)=0$
$\implies x=1,\omega, \omega^2$ (where $\omega$ is a non real cube root of unity)
or
$x^4-4x-1=0$
$\implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1066382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
Trigonometric substitution and Integration of $\frac{1}{x^2\sqrt{x^2+1}} $ Regarding the integral
$$
\int \frac{dx}{x^2\sqrt{x^2 + 1}}
$$
I'm not sure what to do about the extra $x^2$ in the denominator. What can I do about it?
| You might have to use the law of tangent which states that
$$ \sec^2{\theta} =\tan^2{\theta} + 1 $$
or
$$ \sec{\theta} = \sqrt{\tan^2{\theta} + 1} $$
so
$$ \int \frac{dx}{x^2\sqrt{x^2 + 1}}d\theta = \int \frac{d \tan \theta}{\tan^2\sqrt{\tan^2 + 1}}d\theta$$
therefore plugging in sec
$$ \int \frac{d \tan \theta}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
An exercise from Knuth's book - Proving a formula by induction I would like to find a formula for this sum:
$$
\frac{1^3}{1^4+4} - \frac{3^3}{3^4+4} + \frac{5^3}{5^4+4} - ... + \frac{(-1)^n(2n+1)^3}{(2n+1)^4+4}
$$
The answer given (Knuth's book, The Art of Computer Programming Volume 1, Third Edition, Section 1.2.1... | First you have$$ \begin{align} (2n+1)^4+4&= A^4+B^2\\\\ &=(A^4+2A^2B+B^2)-2A^2B\\\\
&=(A^2+B)^2-2A^2B\\\\
&=\left((2n+1)^2+2\right)^2-4\left(2n+1\right)^2\\\\
&=\left((2n+1)^2+2-2(2n+1)\right)\left((2n+1)^2+2+2(2n+1)\right)\\\\
&=\left(4n^2+1\right)\left(4(n+1)^2+1\right) \end{align}$$
Second you have$$ \be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1071024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Show that $f(x,y,z)=0$ if and only if $(\sqrt {x^2+y^2}-1)^2+z^2=r^2$.
Define $f(x,y,z)=(x^2+y^2+r^2-z^2-1)^2-4(x^2+y^2)(r^2-z^2)$, where $0<r<1$ Show that $f(x,y,z)=0$ if and only if $(\sqrt {x^2+y^2}-1)^2+z^2=r^2$.
Here is what I have tried:
Let $f(x,y,z)=0$:
$(x^2+y^2+r^2-z^2-1)^2-4(x^2+y^2)(r^2-z^2)=0$
$x^4+x^2y^... | Choosing good notation is sometimes one of the keys to solving a problem and avoiding complicated mess. This problem is an excellent illustration of that. So let's take $a$ to be $x^2+y^2$ and $b$ to be $r^2-z^2$. The problem reduces down to showing that $$(a+b-1)^2-4ab=0\iff(\sqrt{a}-1)^2-b=0.$$
We have $$\eqalign{(\s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$x^2-y^2=2s$, s cannot be an odd integer How can we prove that if $x^2-y^2=2s$ holds, s cannot be an odd integer. What theorem in number theory should we use?
| $x^2-y^2=(x+y)(x-y)$
Let $y=x+a$ then
$(x+y)(x-y)=(2x+a)(2x-a)=4x^2-a^2$
If $4x^2-a^2=2s$ then $a^2$ must be divisible by 2. Since $a$ is an integer $a$ must be divisible by 2. Thus $a=2b$. Thus $2s=4x^2-4b^2=4c$ where $c=x^2-b^2$ is an integer. Thus $s$ is divisible by $2$.
| {
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"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Evaluating $\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$ Evaluating $$\int{\frac{1}{\sqrt{x^2-1}(x^2+1)}dx}$$ using $ux=\sqrt{x^2-1}$
UPDATE 'official' solution
$$u^2x^2=x^2-1$$
$$x^2=\frac{-1}{u^2-1}$$
$$x^2+1=\frac{u^2-2}{u^2-1}$$
$$2xdx=\frac{-2u}{(u^2-1)^2}$$
$$\int{\frac{x}{x\sqrt{x^2-1}(x^2+1)}dx}$$
$$\int{\frac{... | Let us evaluate the general form of the integral
\begin{align}
\int\frac{\mathrm dx}{(x^2+a^2)\sqrt{x^2-b^2}}&=\int\frac{a\sec^2t}{(a^2\tan^2t+a^2)\sqrt{a^2\tan^2t-b^2}}\mathrm dt\tag1\\[7pt]
&=\frac{1}{a}\int\frac{\cos t}{\sqrt{a^2\sin^2t-b^2\cos^2t}}\mathrm dt\tag2\\[7pt]
&=\frac{1}{a}\int\frac{\cos t}{\sqrt{\left(a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1075675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Liouville sequences We have $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$. We call a finite sequence $a_1,\ldots,a_n$ Liouville if it satisfies \begin{equation}(a_1+\cdots+a_n)^2=a_1^3+\cdots+a_n^3.\end{equation}
Liouville discovered that for all $m\in\mathbb{N}$ the sequence $\tau(d_1),\ldots,\tau(d_n)$ where $d_i$, $i=1,\ldo... | I've searched through sequences of length 4, 5, 6 and 7 with entries up to 10, and there appear to be many other examples. I haven't yet spent time analysing the results to find any patterns, so I'll just list some examples for the moment to answer your question.
Firstly, one obvious observation: sequences which don't... | {
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"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Evaluation of $\int_{0}^{\infty} \cos(x)/(x^2+1)$ using complex analysis. Evaluate:
$$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$
Using only complex analysis.
$$I = \int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = (\frac{1}{2})\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$
Consider a contour $C$ with a upper-ax... | we use $$f\left( z \right) = \frac{{e^{iz} }}{{1 + z^2 }}$$
then take real parts of the resulting integral .using the same contour $C$
\begin{array}{l}
i \in C; - i \notin C \\
{\mathop{\rm Re}\nolimits} s\left( {f;i} \right) = \mathop {\lim }\limits_{z \to i} \left\{ {\left( {z - i} \right)\frac{{e^{iz} }}{{1 + ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Why doesn't squaring the radicand of a square root introduce a plus-minus sign here? The question I have concerns the following problem:
*
*$\sqrt{4x-1} = \sqrt{x+2}-3$
*$(\sqrt{4x-1})^2 = (\sqrt{x+2}-3)^2$
*$\sqrt{4x-1}\times\sqrt{4x-1} = (\sqrt{x+2}-3)\times(\sqrt{x+2}-3)$
*$\sqrt{(4x-1)\times(4x-1)} = (\sqrt{x... | I wonder if the problem could be between line $2$ and line $5$. In line $2$, you square while in line $5$ you take the square root of the square. This could introduce some confusion.
Starting from $$\sqrt{4x-1} = \sqrt{x+2}-3$$ just square to get $$4x-1=(\sqrt{x+2}-3)^2=x+2-6\sqrt{x+2}+9=x+11-6\sqrt{x+2}$$ So, now $$3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1079486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
How to evaluate residue of $\cot z/z^4$ at $z=0$? How to evaluate residue of $\cot z/z^4$ at $z=0$?
As we know :
$$f(x)=f(0)+f'(0)x+f''(0)x^2/2+...$$
but $\cot(0)\to\infty$ or is undefined? I know that:
$$\tan x=x+x^3/3+2x^5/15+...$$
| Well,
\begin{align}\cot z = \frac{\cos z}{\sin z} &= \frac{1 - \frac{z^2}{2!} + \frac{z^4}{4!} + O(z^6)}{z - \frac{z^3}{3!} + \frac{z^5}{5!} + O(z^7)}\\
& = \frac{1}{z}\cdot \left(1 - \frac{z^2}{2!} + \frac{z^4}{4!} + O(z^6)\right)\cdot \left(1 + \frac{z^2}{3!} - \frac{z^4}{5!} +\frac{z^4}{3!3!}+ O(z^6)\right)\\
&= \fr... | {
"language": "en",
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"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Why is $x(\sqrt{x^2+1} - x )$ approaching $1/2$ when $x$ becomes infinite? Why is $$\lim_{x\to \infty} x\left( (x^2+1)^{\frac{1}{2}} - x\right) = \frac{1}{2}?$$
What is the right way to simplify this?
My only idea is: $$x((x²+1)^{\frac{1}{2}} - x) > x(x^{2^{0.5}}) - x^2 = 0$$
But 0 is too imprecise.
| Let $x=\tan\theta$.
Then $L=\displaystyle\lim_{x\to \infty} x( (x^2+1)^{\frac{1}{2}} - x ) \\= \displaystyle\lim_{{\theta\to \frac{\pi}{2}}^{-}} \tan\theta( \sec\theta - \tan\theta )\\=\displaystyle\lim_{{\theta\to \frac{\pi}{2}}^{-}} \frac{\sin\theta-\sin^2\theta}{\cos^2\theta}\\
=\displaystyle\lim_{{\theta\to \fr... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 5
} |
Why do we have $u_n=\frac{1}{\sqrt{n^2-1}}-\frac{1}{\sqrt{n^2+1}}=O(\frac{1}{n^3})$? Why do we have
*
*$u_n=\dfrac{1}{\sqrt{n^2-1}}-\dfrac{1}{\sqrt{n^2+1}}=O\left(\dfrac{1}{n^3}\right)$
*$u_n=e-\left(1+\frac{1}{n}\right)^n\sim \dfrac{e}{2n}$
any help would be appreciated
| For the first one:
$$\begin{align}
\frac{1}{\sqrt{n^2-1}}-\frac{1}{\sqrt{n^2+1}} &= \frac{1}{n}\left(\frac{1}{\sqrt{1-\frac{1}{n^2}}}-\frac{1}{\sqrt{1+\frac{1}{n^2}}}\right) \\
&= \frac{1}{n}\left(\frac{1}{1-\frac{1}{2n^2}+o\!\left(\frac{1}{n^2}\right)}-\frac{1}{1+\frac{1}{2n^2}+o\!\left(\frac{1}{n^2}\right)}\right) \\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$? How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$?
I got answer $\frac{1}{\sqrt{2}}$, but the real answer is $\frac{\sqrt{2}}{2}$. Anyway, calculator for both answers return same numbers.
| Since $2=\sqrt{2}\cdot\sqrt{2}$ you have that $$\frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{\sqrt{2}\sqrt{2}}=\frac{1}{\sqrt{2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1082664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 4
} |
3-variable non-symmetric inequality Prove that for $a,b,c > 0$
$$\frac{a+b-2c}{b+c}+\frac {b+c-2a}{c+a}+\frac {c+a-2b}{a+b} >0$$
What I did is this:- Let $f(a,b,c)=\frac{a+b-2c}{b+c}+\frac {b+c-2a}{c+a}+\frac {c+a-2b}{a+b}$. Therefore
$$\begin{align}f(a,b,c)&=\frac{a+b-2c}{b+c}+\frac {b+c-2a}{c+a}+\frac {c+a-2b}{a+b}\\... | your inequality is equivalent to
$${\frac {{a}^{3}+{a}^{2}b-2\,{a}^{2}c-2\,a{b}^{2}+a{c}^{2}+{b}^{3}+{b}^
{2}c-2\,b{c}^{2}+{c}^{3}}{ \left( b+c \right) \left( c+a \right)
\left( a+b \right) }}
>0$$ and the numerator can written as
$$a(a-c)^2+b(b-a)^2+c(b-c)^2\geq 0$$ and the equal sign holds if $$a=b=c$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1084559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculate limit using L'Hopital Rule -> $ \lim_{x \to 0} \left ( 1 + \frac{1}{x^2} \right )^{x^2} $ I have been trying to solve the following problem, but I seem t have some difficulties.
$$ \lim_{x \to 0} \left ( 1 + \frac{1}{x^2} \right )^{x^2} $$
I tryied to solve it, and the answer I got didn't seem to be right, so... | $$\left(1+\frac{1}{x^2}\right)^{x^2}=\exp\left(x^2\ln\left(1+\frac{1}{x^2}\right)\right)$$
$$x^2\ln\left(1+\frac{1}{x^2}\right)=\frac{\ln\left(1+\frac{1}{x^2}\right)}{\frac{1}{x^2}}$$
therefore
$$\lim_{x\to 0}x^2\ln\left(1+\frac{1}{x^2}\right)=\lim_{x\to 0}\frac{\ln\left(1+\frac{1}{x^2}\right)}{\frac{1}{x^2}}$$
$$\lim_... | {
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"url": "https://math.stackexchange.com/questions/1085527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Inequality $\frac 1{\sqrt{1+xy}}+\frac 1{\sqrt{1+yz}}+\frac 1{\sqrt{1+zx}}\ge \frac 9{\sqrt {10}}$ Let $x,y,z$ be non-negative real numbers such that $x+y+z=1$ , then is the following true? $$\dfrac 1{\sqrt{1+xy}}+\dfrac 1{\sqrt{1+yz}}+\dfrac 1{\sqrt{1+zx}}\ge \dfrac 9{\sqrt {10}}$$
| Here is another approach.
Cauchy-Schwarz says
$$
\begin{align}
(x+y+z)^2
&\le(x^2+y^2+z^2)(1^2+1^2+1^2)\\
&=3(x^2+y^2+z^2)\tag{1}
\end{align}
$$
Therefore,
$$
\begin{align}
xy+yz+zx
&=\tfrac12\left[(x+y+z)^2-\left(x^2+y^2+z^2\right)\right]\\
&\le\tfrac12\left[(x+y+z)^2-\tfrac13(x+y+z)^2\right]\\
&=\tfrac13(x+y+z)^2\tag... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1085590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Let $X$ and $Y$ be i.i.d. $\operatorname{Geom}(p)$, and $N = X + Y$. Find the joint PMF of $X, Y, N$
Let $X$ and $Y$ be i.i.d. $\operatorname{Geom}(p)$, and $N = X + Y$. Find the joint PMF of $X, Y, N$.
I have generally difficulties with such problems, as I get easily confused. Below I detailed my (most probability ... | We have,
$$
P_{X,Y,N}(x,y,n) =
\begin{cases}
P(X=x).P(Y=y) & if x+y=n \\
0 & otherwise
\end{cases}
$$
So,
\begin{align*}
P_{X,Y,N}(x,y,n) = P(X=x, Y=y, N=n)
= P(X=x).P(Y=y)
= pq^x.pq^y
= p^2q^{x+y}
= p^2q^{n}
\end{align*}
Hence, t... | {
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"url": "https://math.stackexchange.com/questions/1085668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How prove $\bigl(\frac{\sin x}{ x}\bigr)^{2} + \frac{\tan x }{ x} >2$ for $0 < x < \frac{\pi}{2}$ How prove $\left(\frac{\sin x}{ x}\right)^{2} + \frac{\tan x }{ x} >2$ for $0 < x < \frac{\pi}{2}$.
Can this be proved with simple way?
| since $\sin(x)>0$ and $x>0$ and $\tan(x)>0$ we have
$$\left(\frac{\sin(x)}{x}\right)^2+\frac{\tan(x)}{x}\geq 2\sqrt{\left(\frac{\sin(x)}{x}\right)^2\cdot \frac{\sin(x)}{x}\cdot\frac{1}{\cos(x)}}$$ Now we have to show that $$\frac{\sin(x)^3}{x^3}>\cos(x)$$ in the given interval, defining $$f(x)=\frac{\sin(x)^3}{x^3}-\co... | {
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"url": "https://math.stackexchange.com/questions/1085780",
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"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Cubic inequality If $x,y,z$ are reals from $[0,1]$, then prove that $$2(x^3+y^3+z^3)-x^2y-y^2z-z^2x \le 3$$ We can assume $x=sin\theta, y=sin\phi,z=sin\gamma$. Therefore inequality can be written as $$\begin{align}&2(sin^3\theta+sin^3\phi+sin^3\gamma)-sin^2\theta.sin\phi-sin^2\phi.sin\gamma-sin^2\gamma.sin\theta\le 3\\... | Assume $x\geq y\geq z$. Then $x^2y\geq y^3, y^2 z\geq z^3,$ and $z^2x\geq z^3$. Consequently
$$
\begin{align*}
2(x^3+y^3+z^3)-x^2y-y^2z-z^2x&\leq 2(x^3+y^3+z^3)-y^3-z^3-z^3\\
&=2x^3+y^3\\
&\leq 3,
\end{align*}
$$
since $x,y\leq 1$.
EDIT: There was some consternation about why one is allowed to assume $x\geq y\geq z$. I... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Solve $x^4+3x^3+6x+4=0$... easier way? So I was playing around with solving polynomials last night and realized that I had no idea how to solve a polynomial with no rational roots, such as $$x^4+3x^3+6x+4=0$$
Using the rational roots test, the possible roots are $\pm1, \pm2, \pm4$, but none of these work.
Because there... | Hint:
First check that $0$ is not a solution, hence $x\neq0\,$, so it is legal to divide by $x^2$. We get
$$
x^2+3x+\frac6x+\frac4{x^2}=0. \tag1
$$
Now note that
$$
\left(x+\frac2x\right)^2=x^2+4+\dfrac{4}{x^2}\iff
x^2+\dfrac{4}{x^2}=\left(x+\dfrac2x\right)^2-4.
$$
So $(1)$ can be written as :
$$
\left(x^2+\dfrac4{x^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 4,
"answer_id": 2
} |
Simple limit problem without L'Hospital's rule $$\lim_{x \to 1}\frac{\sqrt{x^4 + 1} - \sqrt{2}}{\sqrt[3]{x} - 1}$$
We are not supposed to use any derivatives yet, but I can't find any formula that helps here. It's a $\frac{0}{0}$ indeterminate form, and all I think of doing is
$$\frac{\sqrt{x^4 + 1} - \sqrt{2}}{\sqrt[3... | Note that if you put $y=\sqrt[3] x$ then $$\frac {x^4-1}{\sqrt[3]x-1}=\frac {y^{12}-1}{y-1}= y^{11}+y^{10}+\dots +1$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
How to calculate number of triangles and points after dividing a triangle n times? When having a triangle and dividing it n times, how to get the number of triangles and points?
a
/ \
/ \
f-----d
/ \ / \
/ \ / \
c-----e-----b
edit: This triangle is part of an icosahedron as base geometry ... | Each division multiplies the number of small triangles by $4$, and you start with one triangle at $0$ divisions, so the number of triangles after $n$ divisions is $4^n$.
The number of distinct horizontal sides of small triangles after $n$ divisions is
$$\sum_{k=1}^{2^n}k=\frac{2^n(2^n+1)}2=2^{n-1}(2^n+1)\;.$$
There are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1088456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Six of a kind . $$\begin{align}
2^{\color{pink}0}+7^{\color{pink}0}+8^{\color{pink}0}+18^{\color{pink}0}+19^{\color{pink}0}+24^{\color{pink}0}&=3^{\color{pink}0}+4^{\color{pink}0}+12^{\color{pink}0}+14^{\color{pink}0}+22^{\color{pink}0}+23^{\color{pink}0}\\
2^{\color{red}1}+7^{\color{red}1}+8^{\color{red}1}+18^{\color{... | For those who want the quick version, a particular example of Theorem 5 in the link cited by Zander states that if,
$$a^k+b^k+c^k = d^k+e^k+f^k$$
for $k=2,4$, then,
$$\small(x+a)^k+(x+b)^k+(x+c)^k+(x-a)^k+(x-b)^k+(x-c)^k = \\
\small (x+d)^k+(x+e)^k+(x+f)^k+(x-d)^k+(x-e)^k+(x-f)^k$$
for $k=1,2,3,4,5$. The example by the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1090795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 1
} |
Solve logarithmic equation for $x$ to find the inverse of $f(x)= \ln(x+\sqrt{x^2+1})$
Let $f(x)= \ln(x+\sqrt{x^2+1})$. Find $f^{-1}(x)$.
Here is what I got so far: $y= \ln(x+\sqrt{x^2+1})$, rewrite as $x= \ln(y+\sqrt{y^2+1})$,
then $$e^x= y+\sqrt{y^2+1}$$ $$e^x-y= \sqrt{y^2+1}$$ $$ y^2+ e^{2x}-2(e^x)y= 1$$
So if ... | here is a trick in this particular situation. that is to recognize
$$(\sqrt{x^2 + 1} +x)(\sqrt{x^2 + 1} - x) = 1 $$ so that
$$\ln(\sqrt{x^2 + 1} + x) = -\ln(\sqrt{x^2 + 1} - x)$$
now we can find the inverse function. suppose $$\ln(\sqrt{x^2 + 1} + x) = y,$$
then $$\ln(\sqrt{x^2 + 1} - x) = -y$$ exponentiating these t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Solving a differential equation using Laplace transform? $$y''+2y'+ 10 = b\,δ(t-T),\,\begin{cases}y(0)=3\\ y'(0) = 0\end{cases}$$
I managed to solve this equation. My answer is $$y(t) = 3e^{-t} \cos(3t) - e^{-t}\sin(3t)+\dfrac{1}{3}be^{-(t-T)}\sin(3t-3T)u(t-T)$$
I am asked to find values for $b$ and $T$ such that $y(t)... | Cancelling the $e^{-t}$ in $y(t)=0$ for $t>T$ implies that
$$\begin{align*}
3\cos(3t)-\sin(3t)+ \frac{b}{3}e^T\sin 3(t-T)\equiv 0\qquad (1)
\end{align*}$$
Recall the formula (found in most Diff. Eq. books):
$$
A\sin\theta - B\cos\theta = R\sin(\theta-\alpha),
$$
where $R=\sqrt{A^2+B^2}$ and $\alpha=\arcsin\frac{B}{R}$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Integrating factor Can anyone give me some hints as to how to solve the following question?
I have to show that the equation below has an integrating factor of the form $t^2\theta^c$ where $c$ is an integer.
$\theta(\theta^2 + 2t) + 2t(\theta^2+t)\frac{d\theta}{dt} = 0$
I started off by multiplying the integrating fa... | Write your equation as $$t^2\theta^{c+3} + 2t^3\theta^{c+1} + 2t^3\theta^{c+2} \theta'+ 2t^4\theta^c \theta' = \frac 13 (t^3)'\theta^{c+3} + \frac 24 (t^4)'\theta^{c+1} + \frac{2}{c+3} t^3(\theta^{c+3})' + \frac{2}{c+1}t^4(\theta^{c+1})'.$$ Now we want to group the terms with same powers of $t$ and $\theta$. We want ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1092971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solving $\tan x-\tan(2x)=2\sqrt{3}$ $$\tan x-\tan(2x)=2\sqrt{3}$$
TRY #1
$$\begin{align*}
\tan x-\tan(2x)=2\sqrt{3}&\implies\tan x=2\sqrt{3}+\tan{2x}\\
&\implies \tan^2x=\tan^2(2 x)+4 \sqrt{3} \tan(2 x)+12\\
&\implies\tan^2x=(\frac{2\tan x}{1-\tan^2 x})^2+4\sqrt{3}\frac{2\tan x}{1-\tan^2x}+12
\end{align*}$$
but this w... | let $u = \tan x, \tan 2x = \dfrac{2u}{1-u^2}$ your equation becomes $$u - \dfrac{2u}{1-u^2} = 2\sqrt 3 $$ which can be simplified $$f(u) = u^3 - 2\sqrt 3 u^2 + u + 2\sqrt 3 = 0 $$ i don't see any simple roots for this. we know that $f(0) = 2\sqrt 3$ and $f(-1) = -2$ so there is at least one $-1 < u < 0, f(u) = 0$
edi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1093507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Limit of $ \int_{0}^{\frac{\pi}{2}}\sin^n xdx$ and probability I begin with this problem:
Calculate the limit of $\displaystyle \int_{0}^{\frac{\pi}{2}}\sin^n xdx$ when $n\rightarrow \infty$.
It's natural to think of recurrence relation. Let $\displaystyle \int_{0}^{\frac{\pi}{2}}\sin^n xdx=I_n$. By integration by pa... | There is a simple way of proving that the limit is zero. Since:
$$\forall x\in[-\pi/2,\pi/2],\qquad \cos x \leq 1-\frac{4x^2}{\pi^2}, $$
we have:
$$ I_n = \int_{0}^{\pi/2}\sin^n(x)\,dx = \int_{0}^{\pi/2}\cos^n x\,dx \leq \frac{\pi}{2}\int_{0}^{1}(1-x^2)^n\,dx\leq\frac{\pi}{2}\int_{0}^{1}e^{-nx^2}\,dx $$
so:
$$ I_n \leq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1095308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Arithmetic pattern $1 + 2 = 3$, $4 + 5 + 6 = 7 + 8$, and so on I am noticing this pattern:
\begin{align}
1+2&=3\\
4+5+6&=7+8\\
9+10+11+12&=13+14+15 \\
16+17+18+19+20&=21+22+23+24 \\
&\vdots
\end{align}
Is there a general formula that expresses a pattern here, such as for the $n$th term, or the closed form of the entire... | Another way to build up the pattern you see.
Start with the $n$th triangle number, $T_n$ (I'll use $n=5$):
$$1 + 2 + 3 + 4 + 5 = 1 + 2 + 3 + 4 + 5.$$
Add $n^2$ to both sides, except distribute the $n^2$ on one side by adding $n$ to each of the $n$ terms:
$$5^2 + 1 + 2 + 3 + 4 + 5 = 6 + 7 + 8 + 9 + 10.$$
Finally, add $n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1095581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 6
} |
Permutation in discrete math Is the permutation
$$\begin{pmatrix} 1& 2 &3 &4 &5 &6&7 \\ 7 & 4 & 2 & 1 & 3 & 6 & 5 \end{pmatrix}$$
even or odd?
The product of disjoint cycles is $$\begin{pmatrix}1& 7&5&3&2&4\end{pmatrix}\begin{pmatrix}6\end{pmatrix}$$
and the transposition are $$\begin{pmatrix}1 & 7 \end{pmatrix}\begin... | You only need the decomposition in disjoint cycles. Since $\;(1\;7\;5\;3\;2\;4)\;$ as even length, it is an odd cycle and thus is your original permutation
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to show that $B^{-1}\cdot A^{-1}=B\cdot A$? I can't find the solution of this problem:
Given two $n\times n$ square matrices $A,B$ such that $A^2\cdot B^2=I_n$, show that $B^{-1}\cdot A^{-1}=B\cdot A$.
Thanks in advance.
| $A^2 \cdot B^2 = I \Rightarrow A, B$ are invertible. The set of all invertible matrices forms a group. So we also have $B^2 \cdot A^2 = I.$ Hence $B \cdot A = B^{-1}\cdot A^{-1}.$
I'm assuming that the entries of the matrices are from a field.
EDIT: $A^2 \cdot B^2 = I \Rightarrow \text{det}(A^2) \cdot \text{det}(B^2) =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Finding eigenvalues of the linear map of differentiation on polynomial sets I have the following question:
Let $\mathbf{P_2}$ denote the vector space of real polynomials of degree at most two, and let $\mathbf{L}:\mathbf{P_2}\rightarrow\mathbf{P_2}$ be the linear map $\mathbf{L}f=f', $(the derivative of $f$). What are... | Hint: A base for $\mathbf{P_2}$ is $\{1,x,x^2\}$ and the derivatives are
$$1\longmapsto 0,$$
$$x\longmapsto 1,$$
$$x^2\longmapsto 2x.$$
Then the matrix of the transformation is
$\left(\begin{array}{ccc}
0&1&0\\
0&0&2\\
0&0&0
\end{array}\right).$
The characteristic matrix is
$\left(\begin{array}{ccc}
x&-1&0\\
0&x&-2\\
0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1097176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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How do you find that: $x^3+4x^2+x-6=(x+2)(x^2+2x-3)$? How do you find that: $x^3+4x^2+x-6=(x+2)(x^2+2x-3)$?
I get that you can take out the $x$ like so: $x^3+4x^2+x-6=x(x^2+4x+1)-6$ but how do you get the $2$ from here?
| Observe that $x^3 + 4x^2 + x - 6 = x^3 + 4x^2 + 4x - 3x - 6$. Therefore,
\begin{align*}
x^3 + 4x^2 + x - 6 & = x^3 + 4x^2 + 4x - 3x - 6\\
& = x(x^2 + 4x + 4) - 3(x + 2)\\
& = x(x + 2)^2 - 3(x + 2)\\
& = (x + 2)[x(x + 2) - 3)]\\
& = (x + 2)(x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
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Limit of $\left(\frac{x^2+x+1}{2x+1}\right)^{1/(x^2-1)}$ when $x\to1$ $$\displaystyle \lim_{x\to 1}\left(\frac{x^2+x+1}{2x+1}\right)^{\frac{1}{x^2-1}}$$
I know that the two-sided limit of $\frac{1}{x^2-1}$ does not exist. I don't know what to do with $\frac{x^2+x+1}{2x+1}$ to get something else than $1^\infty$
| Hint
Consider $$A=\left(\frac{x^2+x+1}{2x+1}\right)^{\frac{1}{x^2-1}}$$ Taking logarithms $$\log(A)=\frac{1}{x^2-1}\log\left(\frac{x^2+x+1}{2x+1}\right)$$ Now, consider the Taylor series built at $x=1$ $$\frac{x^2+x+1}{2x+1}=1+\frac{x-1}{3}+\frac{1}{9} (x-1)^2+O\left((x-1)^3\right)$$ and use the fact that, for small $y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1099901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Integrate $\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$ using residues I want to find the integral
$$I=\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$$
using contour integration; I am familiar with the trigonometric substitution in real analysis.
There are no branch cuts, there are two second order poles at $z=-1\pm\sqrt{11}i$. I c... | Yes. There is a standard trick for integrating functions over $[0,\infty)$, and that is to exploit the multi-valuedness of the logarithm about a branch cut. Thus, consider
$$\oint_C dz \frac{\log{z}}{(z^2+2 z+12)^2} $$
where $C$ is a keyhole contour as pictured below:
The radius of the small circle is $\epsilon$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
check my work on this problem: given tan(2x), find sin x + cos x? $\tan 2x = - 24/7$
$90^\circ < x < 180^\circ$.
Find the value of $\sin x+\cos x$.
What I have so far:
$\tan(2x) = -\frac{24}{7} \Rightarrow \frac{2\tan(x)}{1-\tan^{2}x} = -\frac{24}{7}$.
Cross-multiplying gives us the quadratic $12\tan^{2}x-7\tan(x)-12 ... | You are very close - basically correct in fact. The only issue I see is that in your calculation of $\sin\arctan\left(-\frac{3}{4}\right)$ and $\cos\arctan\left(-\frac{3}{4}\right)$, you forgot the angles needed to be in the range $90^\circ$ to $180^\circ$, so you need a positive $\sin$ and negative $\cos$. If you just... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Where does $r = 1 + 2\cos(\theta)$ have tangents? Where does:
$$r = 1 + 2\cos(\theta)$$
Have horizontal and vertical tangent lines?
$x = r\cos(\theta) = \cos(\theta) + 2\cos^2(\theta)$
$y = r\sin(\theta) = \sin(\theta) + 2\sin(\theta)\cos(\theta)$
$$dx/d\theta = -\sin(\theta) - 4\sin(\theta)\cos(\theta)$$
$$dy/d\theta ... | You have calculated
$$
\frac{dx}{d\theta}=-\sin\theta(1+4\cos\theta)
$$
and
$$
\frac{dy}{d\theta}=\cos\theta+2\cos2\theta.
$$
Since $\cos2\theta=2\cos^2\theta-1$, the $\frac{dy}{d\theta}$ simplifies to
$$
\frac{dy}{d\theta}=\cos\theta+4\cos^2\theta-2.
$$
Vertical and horizontal tangents
In fact, we do not need to find ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Do I need to use partial fractions to find $\frac{2n+1}{n^2(n+1)^2} = \frac{1}{n^2}-\frac{1}{(n+1)^2}$? I need to simplify $\frac{2n+1}{n^2(n+1^2)}$ as part of an exam question. The solution states $$\frac{2n+1}{n^2(n+1)^2} = \frac{1}{n^2}-\frac{1}{(n+1)^2}$$
In the solution it does not state how this simplification w... | HINT
Use Partial Fraction Decomposition,
$$\frac{2n+1}{n^2(n+1)^2}=\frac An+\frac B{n^2}+\frac C{n+1}+\frac D{(n+1)^2}$$
$$\iff2n+1=An(n+1)^2+B(n+1)^2+Cn^2(n+1)+Dn^2$$
Now compare the constants & the coefficients of $n,n^2,n^3$ to find $A,B,C,D$
Alternatively by observation,
$$\frac{2n+1}{n^2(n+1)^2}=\frac{(n+1)^2-n^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\sum_{k=1}^{m}\frac{1}{k(k+1)}=1-\frac{1}{m+1}$. I know $\sum_{k=1}^{m}\frac{1}{k(k+1)}=1-\frac{1}{m+1}= H_mH_{m+1}$, for $H_m$ the harmonic sum.
I tried many ways to prove it like this
$$\sum_{k=1}^{m}\left(\frac{1}{k(k+1)} + \frac{1}{m+1}\right)=1 $$
$$ \Rightarrow \sum_{k=1}^{m-1} \left(\frac{1}{k(... | Induction would probably work out here too. For $m=1$ we have $$\frac1{1\cdot 2} = 1-\frac12.$$ Then for $m=2$ we have $$\frac{1}{1\cdot2} + \frac{1}{2\cdot 3} = \left(1 - \frac12\right) + \frac1{2\cdot3}=1-\frac{3-1}{2\cdot 3}=1-\frac{1}{3}.$$
Try to do this in general.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Optimize over unit circle to prove $|ax + by| \le \sqrt{a^2 + b^2}$ I have the following problem which, straight off the shelf, seems totally approachable. It's been giving me difficulty however:
Let $a,b,x,y \in \mathbb{R}$, and suppose that $x^2 + y^2 =1$. Prove that $|ax + by| \le \sqrt{a^2 + b^2}$.
I have start... | First of all, there is a fairly easy way to show this inequality:
$$|ax + by| = |(a, b)\cdot (x, y)| \leq \|(a, b)\|\|(x, y)\|\cos\theta \leq \|(a, b)\|\|(x, y)\| = \|(a, b)\| = \sqrt{a^2 + b^2}.$$
As for the Lagrange multipliers method, you're almost there. As $2\lambda x = a$ and $2\lambda y = b$, $x = \frac{a}{2\lam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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Sum of squares of Fibonacci Numbers: $\sum_{i=0}^{n} (F_{2i+1})^2$ $$ \sum_{i=0}^{n} (F_{2i+1})^2 = \;?$$
I know that sum of squares of first $n$ Fibonacci numbers is $F_{n} \times F_{n+1}$.
| We know that:
$$F_n = \frac{1}{\sqrt{5}}\left(\sigma^n - \bar{\sigma}^n\right)\tag{1}$$
where $\sigma,\bar{\sigma}$ are the roots of $x^2-x-1$. On the other hand,
$$ L_n = \sigma^n+\bar{\sigma}^n, \tag{2}$$
and $\sigma\bar{\sigma}=-1$, so:
$$ F_{2n+1}^2 = \frac{1}{5}\left(\sigma^{4n+2}+\bar{\sigma}^{4n+2}+2\right)=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Evaluating the limit $\lim_{n\to\infty} \left[ \frac{n}{n^2+1}+ \frac{n}{n^2+2^2} + \ldots + \frac{n}{n^2+n^2} \right]$ Evaluating the limit $\displaystyle \lim_{n\to\infty} \left[ \frac{n}{n^2+1}+ \frac{n}{n^2+2^2} + \ldots + \frac{n}{n^2+n^2} \right]$
I have a question about the following solution:
We may write it i... | Well you already gave the answer can't figure out what's the problem? You just say: Let $f(x)=\dfrac{1}{1+x^2}$.
$$\lim\limits_{n\to +\infty}\sum\limits_{k=1}^n\frac{n^2}{n^2+k^2}=\lim\limits_{n\to +\infty}\frac{1-0}{n}\sum\limits_{k=1}^nf\left( 0+\frac{1-0}{n}k\right)=\int\limits_0^1f(t)\mathrm dt$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
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Cubes differences and primality In an exercise (Project Euler 131, not to mention it), we are looking for perfect cubes of the form $n^3 + n^2 p$, where p is prime.
I finally got the answer by trial and error but I don't understand why this implies that n and n+p must be perfect cubes. Any proof ?
UPDATE : following up... | We have
$$n^2(n+p)=m^3 \tag{1}$$
1) First let us assume that $p$ does not divide $n$.
1.1) Let $q$ be a prime that divides $n$. We have $\gcd(q,p)=1$ and therefore $\gcd(n+p,q)=1$. If $q^e$ is the highest power of $q$ that divides $n$ then $q^{2e}$ is the highest power of $q$ that divides $m^3$. So $3 \mid e$. This hol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1106180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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All matrices which commute with all $2\times 2$ matrices I would like to find all matrices which commute with all $2\times2$ matrices.
I started solving problem in this way:
1) I have this matrix $A$ with real numbers:
$$A=\left[\begin{array}{cc}a &b\\c &d\end{array}\right]$$
2) Matrix which commute with matrix $A$ ... | we can try four different scenarios with matrices that have 3 zero entires. We can assumed $\mathbf{A}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. We can then try $\mathbf{B}=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$.
Solving for $a,b,c$ & $d$ we get $\mathbf{AB}=\mathbf{BA}$ when $c=0$ and $b=0$.
We can then so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1107111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Implicit Differentiation: $(x/y)+(y/x) =1$ Hi can anyone please tell me where I goes wrong with this question:
Find $ \frac{dy}{dx} $ for the curves defines by this equation:
\begin{align}
\frac{x}{y} + \frac{y}{x} = 1
\end{align}
Here is what I did:
\begin{align}
&\frac{y-xy'}{y^2} + \frac{y'x - y}{x^2} =0 \\... | Note that $y=y(x)$
So we have that
$$\frac{x}{y(x)} + \frac{y(x)}{x} = 1$$
So,
$$\begin{align} 0 &= \frac{d}{dx}\left( \frac{x}{y(x)} + \frac{y(x)}{x}\right) \\
&=\frac{d}{dx}(x) \frac{1}{y(x)}+x \frac{d}{dx}\left(\frac{1}{y(x)}\right) \\
&= \frac{1}{y(x)}+ x \ \frac{-1}{y^2(x)} \frac{dy}{dx} \\
&= \frac{1}{y(x)}\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1107869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
} |
$\sum _{n=1}^{\infty } \frac{1}{n (n+1) (n+2)}$ Understand the representation $$\sum _{n=1}^{\infty } \frac{1}{n (n+1) (n+2)}$$=$$\frac{1}{2} \left(-\frac{2}{n+1}+\frac{1}{n+2}+\frac{1}{n}\right)$$
$$s_n=\frac{1}{2} \left(\left(\frac{1}{n+2}+\frac{1}{n}-\frac{2}{n+1}\right)+\left(\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}... | You should write: (define $s_n$ as the sequence of partial sums of the series)
$s_n=\sum _{k=1}^{n} \frac{1}{k (k+1) (k+2)}$
$=\sum _{k=1}^{n} {\frac{1}{2} \left(-\frac{2}{k+1}+\frac{1}{k+2}+\frac{1}{k}\right)}$
$=\frac{1}{2}\left(-2\sum_{k=1}^{n}\frac{1}{k+1}+\sum_{k=1}^{n}\frac{1}{k+2}+\sum_{k=1}^{n}\frac{1}{k}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1108626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Prove or disprove: $\sum a_n$ convergent, where $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$.
Let $a_n=2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}$. Show that $a_n>0\ \forall\ n\ge1$.
Prove or disprove: $\sum\limits_{n=1}^\infty a_n$ is convergent.
I can't show that $a_n > 0\ \forall n\ge1$. I tried using induction but it wouldn... | You don't have to see the formula. The $n$th term of the sequence is of the form $f(n+1) - 2f(n) + f(n-1)$, where $f(n) = -\sqrt{n}$.
This is a "difference of differences"; $f(n+1) - 2f(n) + f(n-1) = g(n+1) - g(n)$, where $g(n) = f(n) - f(n-1)$. So the infinite sum is telescoping and will equal $\lim_{n \rightarrow \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1109867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 3
} |
Prove that $1\cdot f(1)+ 2\cdot f(2)+ ...+ n\cdot f(n) \leq n(n+1)(2n+1)/6$ where $f(n+1) = f(f(n))+1$ Consider checking function $\mathbb{N}\to \mathbb{N}$ relationship $f(n+1) = f(f(n))+1$, for any positive integer $n$.
Prove that $1\cdot f(1)+ 2\cdot f(2)+ ...+ n\cdot f(n) \leq n(n+1)(2n+1)/6$ for any positive inte... | Since $1\cdot 1+ 2\cdot 2+ \cdots + n\cdot n = n(n+1)(2n+1)/6$, it is enough to prove that $f(n)\le n$, for all $n$.
We shall prove that $f(n)\le n$ for all $n$, by full induction.
Assume $f(m)\le m$ for all $m\le n$. Let's prove that $f(n+1) \le n+1$.
Let $m=f(n) \le n$. Then $f(f(n))=f(m) \le m=f(n)$.
So, $f(n+1) = f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1113517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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How to find all integer solutions of $p^2+q^2=((2q+1)^2+q+1)^2+1$ $$p^2+q^2=((2q+1)^2+q+1)^2+1$$
How do I find integer solutions to this equation? I've already found $(p,q)=(11,1)$. How do I go about finding new ones?
| $$\begin{align}p^2+q^2&=((2q+1)^2+q+1)^2+1\\\implies p^2+q^2&=16q^4+40q^3+41q^2+20q+5 \\\implies p^2&=16q^4+40q^3+40q^2+20q+5\end{align}$$
So for an integer $q$ if $16q^4+40q^3+40q^2+20q+5$ is a perfect square, then you will get integer solutions for your equation.
*
*$q=1\implies16q^4+40q^3+40q^2+20q+5=121=11^2$
*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1115259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Measure of a modified Cantor set Suppose a modified Cantor set:
Starting with $E_0 = [0,1]$, we delete the middle interval of length $1/3$, then we delete the middle intervals of length $1/15$, and so on; in each step we delete from all the intervals of set $E_m$ the middle of length
$$\frac{1}{3\cdot5\cdot\ \space ..... | How about this:
$$\sum_{n=0}^\infty \frac{2^n}{3 \cdot 5 \cdot \ldots \cdot (2n+3)} \ge 1/ 3 $$
For the opposite direction:
$$\sum_{n=0}^\infty \frac{2^n}{3 \cdot 5 \cdot \ldots \cdot (2n+3)} \le \sum_{n=0}^\infty \frac{2^n}{2 \cdot 4 \cdot \ldots \cdot (2n+2)} = \sum_{n=0}^\infty \frac{2^n}{2^{n+1} (n+1)!} = \frac 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1118171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How can we calculate the exponential form of a rotation matrix Considering the rotation matrix:
$$
A(\theta) = \left( \begin{array}{cc}
\cos\space\theta & -\sin \space\theta \\
\sin \space\theta & \cos\space\theta \\
\end{array} \right)
$$
How can I calculate $(A(\theta))^n$ where n ≥ 1 ? I'm not sure what to do nor h... | $A(\theta)$ is called the rotation matrix simply because it rotates a point in the plane by an angle $\theta$. What happens when I apply this matrix $n$ times? I simply rotate the point $n$ times. How much has it rotated in the end? By an angle $n\cdot\theta$. If you can get this sort of intuitive grasp, you can say th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1119548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Arc length contest! Minimize the arc length of $f(x)$ when given three conditions. Contest: Give an example of a continuous function $f$ that satisfies three conditions:
*
*$f(x) \geq 0$ on the interval $0\leq x\leq 1$;
*$f(0)=0$ and $f(1)=0$;
*the area bounded by the graph of $f$ and the $x$-axis between $x=0$ an... | Consider the following:
$$f(x)=\begin{cases}
cx & \text{ if } 0 \le x \le \frac{1}{c} \\
-cx+c & \text{ if } 1-c < x \le 1 \\
1 & \text{ otherwise } \\
\end{cases}$$
If we take $c\to\infty$ we get that it is an arc length of 3.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1122929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "65",
"answer_count": 12,
"answer_id": 4
} |
Find all primes $p$ such that $p^2-p+1$ is a perfect cube Find all primes $p$ such that $p^2-p+1$ is a perfect cube.
I found out that p is of the form $18n+1$ and $p=19$ is a solution but I am not getting anything further.
$p^2-p-(m^3-1)=0$
$1+4(m^3-1)=k^2$
$4m^3-3=k^2$
every square is $0,1,4 or 7(\mod9)$ and every cub... | this is an easier answer:
$$$$we have$$p^2-p=p(p-1)=m^3-1=(m-1)(m^2+m+1)$$we can see $p>m-1$ so we have $$p-1|m-1\to p=1\, MODm-1$$and we have$$m^2+m+1=3\, MODm-1$$we know $$p|m^2+m+1$$ so we have$$m^2+m+1=3p\, or\, m^2+m+1\ge (m+2)p$$if $$m^2+m+1=3p$$ we can see p=19,we have $3(m-1)=p-1$ so we have $m^2+m+1=9m-6$ so m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1123011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Use induction to prove that sum of the first $2n$ terms of the series $1^2-3^2+5^2-7^2+…$ is $-8n^2$. Use induction to prove that sum of the first $2n$ terms of the series $1^2-3^2+5^2-7^2+…$ is $-8n^2$.
I came up with the formula $\displaystyle\sum_{r=1}^{2n} (-1)^{r+1}(2r-1)^2=-8n^2$ but I got stuck proving it by ind... | For $n=1$ we have:
$$1^2 - 3^2 = -8 = -8\cdot1^2$$
Now assume that the statement holds for some n. Then we have a sum:
$$1^2-3^2+...+ (4n-3)^2 - (4n - 1)^2= -8n^2$$
Adding the next two terms to both sides we get:
\begin{align}
&1^2-3^2+...+ (4n-3)^2 - (4n - 1)^2 + (4n + 1)^2 - (4n + 3)^2 \\
&= -8n^2 + (4n + 1)^2 - (4n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1123189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Parametrization of solutions of diophantine equation $x^2 + y^2 = z^2 + w^2$ I need integer solutions of $x^2 + y^2 = z^2 + w^2$ parametrized. Can it be done? Thanks.
| \begin{align*}
\left(\dfrac{\alpha^2-\beta^2+\gamma^2-\delta^2}{\alpha^2+\beta^2-\gamma^2-\delta^2}\right)^2+\left(\dfrac{2\left(\alpha\beta-\gamma\delta\right)}{\alpha^2+\beta^2-\gamma^2-\delta^2}\right)^2-\left(\dfrac{2\left(\alpha\gamma-\beta\delta\right)}{\alpha^2+\beta^2-\gamma^2-\delta^2}\right)^2=\large{1}
\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1126517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Evaluating $\int\frac{\mathrm dx}{\sqrt{\lfloor 1+ \sqrt{1+x}\rfloor}}$ How can I solve this integral?
$$\int\frac{\mathrm dx}{\sqrt{\lfloor 1+ \sqrt{1+x}\rfloor}}$$
| Notice for any integer $n \ge 2$ and $x \ge 0$,
$$\begin{align}
\left\lfloor 1 + \sqrt{1+x}\right\rfloor = n
&\iff n \le 1 + \sqrt{1+x} < n+1\\
&\iff x \in [ (n-1)^2 - 1, n^2 - 1 )
\end{align}$$
To perform the integral over some interval $[0,X]$ where $X > 0$, you just need to split the interval $[0,X]$ to intervals of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1127465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Show that $f(x) = \frac{x^3}{1+x^2}$ is bijective This seems like a simple question, but I'm stuck: how do I show that $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as $f(x) = \frac{x^3}{1+x^2}$ is bijective?
I want to demonstrate that it is both injective and surjective. To show that it's injective, I need to show th... | you don't need calculus to show that $f(x) = \dfrac{x^3}{1+x^2}$ is 1-1.
suppose $$\dfrac{a^3}{1+a^2} = \dfrac{b^3}{1+b^2} \tag 1$$ we will show that this implies $a = b$ proving $f$ is 1-1.
cleaning up $(1)$ gives $$(a-b)(a^2 + ab + b^2 + a^2b^2) = 0$$ now use the fact that $a^2 + ab + b^2 > 0$ for $a \neq 0, b \neq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Integration problem: $\int x^{2} -x 4^{-x^{2}} dx$ I need to integrate $$\int x^{2} -x 4^{-x^{2}} dx$$
and I know the answer I got is wrong. However, I can't figure out where I went wrong. These steps I took:
Appreciate any help!
| $$\begin{align}
\int x^{2} - x4^{-x^{2}} dx &= \int x^{2} dx - \int x4^{-x^{2}} dx \\
&= \frac{x^{3}}{3} + \int -x4^{-x^{2}} dx \\
\end{align}$$
notice that
$$\frac{d}{dx} 4^{-x^{2}} = -2x\cdot \ln4 \cdot 4^{-x^{2}}$$
So if
$$\begin{align}
u &= 4^{-x^{2}} \\
\implies du &= -2x\cdot \ln4 \cdot 4^{-x^{2}} dx \\
\implie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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What does it mean to solve or find solutions in mathematics? Something that has been really confusing me lately is that this equation has four solutions $$3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)$$
But what does that mean? Until now solutions to me has meant, what are the coordinates of $x$ when $y$ equals a given value, normal... | Definition: Solve. We say an equation is solved if and only if we have listed the set of all objects which make the equation true. We say we have found a solution if we have found an element of the solution set.
Example: If $x= 0$, then
\begin{align}
3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)
\end{align}
becomes
\begin{align}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1133253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 3
} |
Series approximation to $\int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du$ I have figured out by graphing that, for small $x$:
$$
\int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du\approx\log(1/x)+\pi/2+O(x)
$$
However, I am unable to prove that this is the case.
As $x\to 0$ the integral diverges at $u=1$. We can chec... | I think the resulting behavior is indeed a log as $x \to 0$. To see why, note that the denominator may be written as
$$f(u,x) = 2 u^3 - 3 u^2 + 1 + 2 x (1 - u^2)$$
This implies that $1-u$ is a factor of $f(u,x)$ independent of $x$. In fact,
$$f(u,x) = (1-u)(1+2 x + (1+2 x) u - 2 u^2) $$
The zeroes of the quadratic ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1133695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Find all continuous functions satisfying $\int_0^xf=(f(x))^2+C$ for some constant $C \neq 0$. Find all continuous functions $f$ satisfying
$$\int_0^xf=(f(x))^2+C$$ for some constant $C \neq 0$, assuming that $f$ has at most one $0$.
I have a question about the solution to this problem. It says that
Clearly $f^2$ is d... | putting $x = 0,$ gives $C = -f^2(0)$ differentiating gives you $f = 2ff'$ which has solution $f = 0$ which implies $C = 0$ or $f = \frac{x}{2} + f(0).$
$\int_0^x(f(0) + t/2) \, dt = xf(0) + \frac{x^2}{4}$ and $f^2(x) + C = \frac{x^2}{4} + xf(0) + f^2(0) - f^2(0) = xf(0) + \frac{x^2}{4}$ it verifies that
$$f(x) = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
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Remainder when $x^{1000000}$ is divided by $x^3 + x +1$ in $\mathbb{Z}_2[x]$ I tried the traditional algorithm of long division hoping to find a pattern, but I was not able to.
I then tried using the root of $x^3 + x + 1$ $\left(x \sim -0.7\right)$ in the equation:
$$x^{1000000} = \left(x^3 + x + 1\right) q\left(x\ri... | Just look at the low powers of $x$. Eventually they will cycle.
$x^0 = 1$
$x, x^2$ cannot be simplified, but
$x^3 = 0 - (x+1) = -x-1 = x+1$
$x^4 = x(x+1) = x^2+x$
$x^5 = x(x^2+x) = x^3+x^2 = x+1+x^2 = x^2+x+1$
$x^6 = x(x^2+x+1) = x^3+x^2+x = x+1+x^2+x = x^2+1$
$x^7 = x(x^2+1) = x^3+x = x+1+x = 1$.
So the powers of $x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Unknown Taylor expansion I have come across a few apparently related Taylor expansions, as detailed below:
\begin{align}
&\dots\frac{a^7}{140}-\frac{a^6}{80}-\frac{3 a^5}{40}-\frac{a^4}{8}+\frac{a^2}{2}+a+1&&=\exp \left(a-\frac{a^3}{6}\right)&\tag{1}\\
\\
&\dots\frac{11 a^7 b}{1120}+\frac{a^6 b}{32}+\frac{a^5 b}{40}-\f... | $$
\frac{1}{2} (1-a)\exp\left(a-\frac{a^3}{6}\right)=
\frac{1}{2}-\frac{a^2}{4}-\frac{a^3}{4}-\frac{a^4}{16}+\frac{a^5}{40}+\frac{a^6}{32}+\frac{11 a^7}{1120}-\frac{a^8}{1280}-\frac{43
a^9}{20160}-\frac{69 a^{10}}{89600}+O\left(a^{11}\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1136848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Integrating $ \int \frac{1}{\sqrt{x^2+4}}\,dx $ using Trigonometric Substitution I'm reviewing integration by trigonometric substitution in anticipation of covering it in class next week. I seem to be a bit rusty and keep catching myself making various mistakes. On this particular problem I keep getting the same answ... | You are correct still. Notice that
$$
\begin{align}
\ln\Bigg|\frac{\sqrt{x^2+4}}{2} + \frac{x}{2}\Bigg| + C &= \ln\Bigg|\frac{\sqrt{x^2+4}+x}{2}\Bigg| + C\\
&= \ln\Bigg|\sqrt{x^2+4}+x\Bigg|-\ln(2) + C\\
&= \ln\Bigg|\sqrt{x^2+4}+x\Bigg|+C'
\end{align}
$$
where $C'$ is still an arbitrary constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1137040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Calculate the trace of all elements in $F_8$ I got the following exercise where you have to calc the trace of all elements in ${F_8}$ which is constructed as ${F_2}[x]$/(${x^3+x+1}$)${F_2}[x]$.
Up to now I did those steps:
1) Find all elements in ${F_8}$ which are in my opinion: $0,1,x,x+1,x^2,x^2+1,x^2+x,x^2+x+1$
2) T... | We have here an extension of dimension three of the prime field $\;\Bbb F_2\;$ , and in fact
$$\Bbb F_{2^3}=\Bbb F_8\cong\Bbb F_2[x]/\langle x^3+x+1\rangle = \text{Span}_{\Bbb F_2}\{\overline 1,\overline x,\overline{x^2}\}\;$$
where for $\;v\in\Bbb F_2[x]\;,\;\;\overline v:=v+\langle x^3+x+1\rangle\in\Bbb F_2[x]/\lang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1137708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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When $\sqrt{(x+a)^2 -b}$ is an integer? While working on integer factorization problem, I came to this:
How to find for which values of $x$ the next equation is an integer?
$$\sqrt{(x+a)^2 -b}$$
*
*$a,b$ are positive known integers
In order to find a solution to this question, should I test all the values of $x$? ... | Let that integer be $y^2$ then $b=(x+a)^2-y^2=(x+a+y)(x+a-y)$. So, you only need to consider factorizations of $b=b_1b_2$ and the corresponding solutions of the system $$\begin{align}x+a+y&=b_1\\x+a-y&=b_2\end{align}$$
Example:
Take $a=0$, $b=3$. The factorizations of $b=3$ are $$\begin{align}1&\cdot3\\3&\cdot1\\(-1)&\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1139447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Finding $\lim _{n\to \infty}\frac{\sqrt{n+\sqrt{n^2+1}}}{\sqrt{3n+1}}$
Find the limit: $\displaystyle\lim _{n\to \infty}\frac{\sqrt{n+\sqrt{n^2+1}}}{\sqrt{3n+1}}$
My attempt:
$\begin {align}\displaystyle\lim _{n\to \infty}\frac{\sqrt{n+\sqrt{n^2+1}}}{\sqrt{3n+1}} &=
\lim _{n\to \infty}\frac{\sqrt{n+n^4\sqrt{1+\frac{... | $$\lim_{n\to\infty}\frac{n+\sqrt{n^2+1}}{3n+1}=\lim_{n\to\infty}\frac{1+\sqrt{1+1/n^2}}{3+1/n}$$
$$=\frac{1+\sqrt1}3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1140469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
summation of a trigonometric series How to evaluate $\tan^2(1) + \tan^2(3) + \tan^2(5) + \tan^2(7) + \ldots + \tan^2(89)$?
Angles are given in degrees.
I tried converting $\tan(89)$ as $\cot(1)$ and then tried combining $\tan(1)$ and $\cot(1)$, but later got stuck.
| For integer $n,\tan(2n+1)90^\circ=\tan90^\circ=\infty$
If $\tan90x=\infty,90x=180^\circ m+90^\circ=90^\circ(2m+1)$ where $m$ is any integer
$\implies x=(2m+1)^\circ$ where $0\le m\le89$
Like Trig sum: $\tan ^21^\circ+\tan ^22^\circ+...+\tan^2 89^\circ = ?$,
$$\tan90x=\frac{\binom{90}1t-\binom{90}3t^3+\cdots+\binom{90}... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 1
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How can I show that $\prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right) = \frac{\left(-1\right)^{k-1}}{2}$? Assume $k$ positive integer. How can I show that
$$ \tag 1 \prod_{{n\geq1,\, n\neq k}} \left(1-\frac{k^{2}}{n^{2}}\right)
= \frac{\left(-1\right)^{k-1}}{2}? $$
I know that
$$ \tag 2 \underset{n\g... | If $N \ge k+1$, then
\begin{align}
\prod_{\underset{n\neq k}{n = 1}}^N \left(1 - \frac{k^2}{n^2}\right)&= \prod_{n = 1}^{k-1}\left(1 - \frac{k^2}{n^2}\right)\prod_{n =k+ 1}^N \left(1 - \frac{k^2}{n^2}\right)\\
&= \prod_{n = 1}^{k-1} \frac{(n-k)(n+k)}{n^2} \prod_{n = k+1}^N \frac{(n-k)(n+k)}{n^2}\\
&= \frac{(-1)^{k-1}(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1142703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to prove that $\cot{\frac{11\pi}{18}} +\cot{\frac{2\pi}{9}}=4\sin{\frac{\pi}{18}}\cot{\frac{2\pi}{9}}$ How to prove this trigonometric identities ?
$$\cot{\frac{11\pi}{18}} +\cot{\frac{2\pi}{9}}=4\sin{\frac{\pi}{18}}\cot{\frac{2\pi}{9}}$$
Thank you.
| We have to check:
$$ -\tan\frac{\pi}{9}+\cot\frac{2\pi}{9}=4\cos\frac{4\pi}{9}\cot\frac{2\pi}{9} $$
hence, by multiplying both sides by $\sin\frac{2\pi}{9}=2\sin\frac{\pi}{9}\cos\frac{\pi}{9}$:
$$ -2\sin^2\frac{\pi}{9}+\cos\frac{2\pi}{9} = 4\cos\frac{4\pi}{9}\cos\frac{2\pi}{9} $$
so we just have to check that $x=\cos\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1146390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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How to calculate: $\lim \limits_{x \to 0}$ $\frac{-1}{\sin^2x}$ + $\frac{1}{x^2}$ How to calculate: $$\lim \limits_{x \to 0}\frac{1}{x^2} - \frac{1}{\sin^2x} $$
any suggestions what I can do here?
| $$-\frac{1}{\sin(x)^2} + \frac{1}{x^2} = \frac{-x^2+\sin(x)^2}{x^2 \sin(x)^2}.$$
You can use L'Hopital's rule now, but it will take multiple applications since both the numerator and denominator are very small (of order $x^4$, it turns out) near $0$.
A nicer approach is to use the Maclaurin series for $\sin$ to see tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1147180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Induction exercise check-up Prove by induction on $n$ that $13$ divides $2^{4n+2} + 3^{n+2}$ for all natural $n$.
For base case it is divisble by 13, and
$2^{4n+6} + 3^{n+3}$ must be divisble too.
$16 * 2^{4n+2}+ 3* 3^{n+2}$
If we denote $2^{4n+2} + 3^{n+2}$ as some $13y$,
we have $3*13y + 13*2^{4n+2}$
Didn't I make ... | your assumption is necessary in the induction
the steps of induction is :
1. calculate when n = 1 it is true or not
2. assume when n = k is true then check whether n = k + 1 is also true.
*
*for n = 1,
$2^{4n+2}+3^{n+2} = 2^6+3^3 = 7*13$
*assume $2^{4k+2}+3^{k+2}$ is divisible then
$2^{4(k+1)+2}+3^{(k+1)+2}$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1147659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
How to prove $\sum_{k=0}^{\infty}k^2x^{k} = \frac{x(1+x)}{(1-x)^3}\text{, }|x| < 1$? How do I prove that the summation $$\sum_{k=0}^{\infty}k^2x^{k} = \dfrac{x(1+x)}{(1-x)^3}\text{, }|x| < 1\text{?}$$
| Write
$$
\frac{x(1+x)}{(1-x)^3}=\frac{x^2-2x+1+3x-1}{(1-x)^3}=\frac{1}{1-x}+3x\frac{1}{(1-x)^3}-\frac{1}{(1-x)^3}.
$$
Let's begin with
$$
\frac{1}{1-x}=\sum_{k=0}^\infty x^k.
$$
Differentiate $2$ times, which gives
$$
\frac{2}{(1-x)^3}=\sum_{k=2}^\infty k(k-1)x^{k-2}=\sum_{k=0}^\infty(k+2)(k+1)x^k.
$$
Thus, your RHS eq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1147987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Rationalize the denominator: $(\frac{3}{2x^2}) ^{1/4}$ My answer is $\frac{(6x^2)^{1/4}}{ 2x^2}$
However the book's answer is $\frac{(24x^2)^{1/4}}{2x}$
Where did the book get that from?
Here's my work:
$$\sqrt[4]{\frac{3}{2x^2}}= \frac{\sqrt[4]{3}}{\sqrt[4]{2x^2}}\cdot \frac{\sqrt[4]{2x^2}}{\sqrt[4]{2x^2}}$$
$$= \... | Your error is when you go from the first line to the second. We have
$$\sqrt[4]{\frac{3}{2x^2}}= \frac{\sqrt[4]{3}}{\sqrt[4]{2x^2}}\cdot \frac{\sqrt[4]{2x^2}}{\sqrt[4]{2x^2}} = \frac{\sqrt[4]{6x^2}}{\sqrt[4]{4x^4}}$$
You can re-express the denominator as $\sqrt[4]{4}\sqrt[4]{x^4} = |x|\sqrt[4]{4}$, which is almost fre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1148309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Factoring Polynomials (x^4) - using completing squares
Exercise 6. By viewing the polynomials as a difference of two squares, factorise the following polymomials.
*
*$x^4+x^2+1$,
*$x^4+3x^2+4$,
*$x^4+4$.
To solve difference of two squares isnt it necessary to have negatives? im confused about how t... | *
*$x^4+x^2+1= (x^2)^2+(x^2)\cdot 1+1^2=(x^2)^2+2\cdot (x^2)\cdot 1+1^2- (x^2)\cdot 1=(x^2+1)^2-x^2=(x^2+1-x)(x^2+1+x)=(x^2+x+1)(x^2-x+1).$
*$x^4+3x^2+4= (x^2)^2+3\cdot (x^2)+2^2=(x^2)^2+4\cdot (x^2)+2^2-x^2=(x^2)^2+2\cdot 2\cdot x^2+2^2-x^2=(x^2+2)^2-x^2=(x^2+2-x)(x^2+2+x)=(x^2+x+2)(x^2-x+2).$
*$x^4+4=(x^2)^2+2^2 =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1149290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove that $\sqrt{a^2+(1-b)^2}+\sqrt{b^2+(1-c)^2}+\sqrt{c^2+(1-a)^2}\geq\frac{3\sqrt{2}}{2}$
Prove that the following inequality
$$\sqrt{a^2+(1-b)^2}+\sqrt{b^2+(1-c)^2}+\sqrt{c^2+(1-a)^2}\geq\frac{3\sqrt{2}}{2}$$
holds for arbitrary real numbers $a$, $b$ and $c.$
Someone says, "It's very easy problem. It can also... | Hint:
$$\sqrt{\frac{a^2+(1-b)^2}{2}}\ge \frac{|a| + |1-b|}{2}$$
Add up, RHS $\ge \frac{|a|+|1-a|+|b|+|1-b|+|c|+|1-c|}{2}\ge \frac{3}{2}$, on LHS it's our sum divided by $\sqrt{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1149980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"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.