Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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How do we express the equation corresponding to the intersection of the planes $x + y + z = 1$ and $x + 2y + 2z = 0$? If a system with three unknowns and two equations are such that
$$
\begin{align}
x+y+z=1&\\
x+2y+2z=0
\end{align}
$$
In the answer it says that this system can be represented as
$$
\begin{pmatrix}
x \\ ... | The proposed system of equations corresponds to the intersection of two planes which are not parallel, also known as a line. In order to obtain one of its possible parametrizations, subtract the first equation from the second in order to obtain:
\begin{align*}
(x + 2y + 2z) - (x + y + z) = 0 - 1 & \Longleftrightarrow y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4600941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why are there complex numbers in the exact solution of $\sum_{n=1}^{\infty} \dfrac{1}{n^4+1}$? Knowing that
\begin{align}
\cot(z)=\frac{1}{z}-2z\cdot\sum_{n=1}^{\infty} \dfrac{1}{\pi^2n^2-z^2}
\end{align}
we can easily calculate the value of
\begin{align}
\sum_{n=1}^{\infty} \dfrac{1}{n^2+1}
\end{align}
by just plugg... | Define $t=\pi\sqrt{i}$ and $f(z):=z\cot(z)$.
Now $f(-z)=f(z)$ for all $z$ so $f(z)=g(z^2).$
Your result is
$$ S:=\sum_{n=1}^\infty\frac1{n^4+1} = \frac14\big[-2+f(t)+f(it)\big]
= -1/2 + (g(i\pi^2)+g(-i\pi^2))/4.$$
Rewrite this as $ S = -1/2 + \Re g(i\pi^2)/2 \approx 0.57848 $ where
$\Re z$ is the real part of $z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4602781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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For integer $x \ge 1$, does it follow that $\left(\frac{x^2+2x+1}{x^2+2x}\right)^{x+1} > \frac{x+2}{x+1}$ Would I be correct that the answer here is yes?
Here is my thinking:
*
*For $x \ge 1$, $\dfrac{x+2}{x+1}$ is strictly decreasing.
*For $x \ge 1$, $\left(\dfrac{x^2 + 2x + 1}{x^2 + 2x}\right)^{x+1}$ is strictly i... | As commenters have pointed out, the left side is not increasing, but decreasing.
You can see this by showing:
$$\left(\frac{x^2+2x+1}{x^2+2x}\right)^{(x+1)^2}\to e,$$ so the left side of your inequality is in the range of $e^{1/(x+1)},$ and converges to $1.$
Bernoulli's inequality says that if $y\geq -1$ and $n$ is a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4607355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Solve the equation $\log_x(9x^2)\cdot\log_3^2(x)=4$ Solve the equation $$\log_x(9x^2)\cdot\log_3^2(x)=4$$
The answers are $x_1=\dfrac19$ and $x_2=3$.
For the range we have: $$D_x:\begin{cases}x>0\\x\ne1\\9x^2>0\iff x\ne0\end{cases}\iff x\in(0;1)\cup(1;+\infty)$$
I wrote the equation as follows using $\log_a(b)=\dfra... | You are on the right track!
Based on the assumption that $x\in(0,1)\cup(1,+\infty)$, we can rewrite the first term from the LHS of the original equation as follows:
\begin{align*}
\log_{x}(9x^{2}) = \log_{x}(9) + 2\log_{x}(x) = \log_{x}(3^{2}) + 2 = 2\log_{x}(3) + 2
\end{align*}
As you have already noticed, the followi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4608594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Evaluate $\int \frac{\sqrt{1+x^4}}{1-x^4}dx$ How to evaluate
$$\int \frac{\sqrt{1+x^4}}{1-x^4}dx.$$
If we substitute $x=\sqrt{\tan{\theta}}$, it becomes
$$\int \frac{1}{2\cos{\theta}\cos{2\theta}\sqrt{\tan{\theta}}}d\theta.$$
What can I do next ?
Edit
Are these steps correct ?
$$=\int \frac{1}{2\cos{\theta}\cos{2\the... | Alternatively, substitute $t=\frac{\sqrt2 x}{\sqrt{1+x^4}}$. Then
$$\frac{\sqrt{1+x^4}}{1-x^4}=\frac{xt}{\sqrt2(t^2-x^2)}, \>\>\>\>\>dx =\frac{x}{t(1-t^2x^2)}dt$$
and
\begin{align}
&\int \frac{\sqrt{1+x^4}}{1-x^4}dx\\
=&\ \frac1{\sqrt2}\int \frac{x^2}{(t^2-x^2)(1-t^2 x^2)}dt
= \frac1{\sqrt2}\int \frac1{t^2(\frac1{x^2}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4609026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Evaluate the limit $\lim_{n \rightarrow \infty}\sum_{k=1}^n\frac{1}{x_{1}^{2}x_{2}^2...x_{k}^2}$ We have the sequences $(x_{n})_{n\geq1}$$(y_{n})_{n\geq1}$ with positive real numbers. $x_{1}=\sqrt{2}$, $y_{1}=1$ and $y_{n}=y_{n-1}\cdot x_{n}^{2}-3$ for every $n\geq2$. We know the sequence $(y_{n})_{n\geq1}$ is bounded.... | Introduce an auxilary sequence $z_n = \frac{y_n}{y_n+3} \iff y_n = 3\frac{z_n}{1-z_n}$. For $i \ge 2$, we have
$$y_i = y_{i-1}x_i^2 -3
\implies \frac{1}{x_i^2} = \frac{y_{i-1}}{y_i+3} =
\frac{3\frac{z_{i-1}}{1-z_{i-1}}}{
3\frac{z_i}{1-z_i} + 3} = z_{i-1}\frac{1-z_i}{1-z_{i-1}}
$$
Multiply from $i = 2$ to any $k \ge 2$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4611635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How do we prove $x^6+x^5+4x^4-12x^3+4x^2+x+1\geq 0$? Question
How do we prove the following for all $x \in \mathbb{R}$ :
$$x^6+x^5+4x^4-12x^3+4x^2+x+1\geq 0 $$
My Progress
We can factorise the left hand side of the desired inequality as follows:
$$x^6+x^5+4x^4-12x^3+4x^2+x+1=(x-1)^2(x^4+3x^3+9x^2+3x+1)$$
However, after... | Let $P(x) = x^4 + 3x^3 + 9x^2 + 3x + 1$
$P(x≥0) \;≥\; 1 \;>\; 0\;$
$P(y=-x) \;=\; y^4 - 3y^3 + 9y^2 - 3y + 1 = (y-1)^4 + (y^3+3y^2+y)$
$P(x<0 \;⇒\;y>0) \;>\; 0 \quad\quad $ // RHS first term ≥ 0, second term > 0
$\forall x\in\mathbb {R} \;⇒\; P(x)>0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4611737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 6
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Find $\frac {dy}{dx}$ if $y=\frac{x\sin^{-1}x}{\sqrt{1-x^2}}$ taking JDs advice i used $(fg)'=f'g+fg'$ rule
$$f=\frac x{\sqrt{(1-x^2)}}$$ $$f'=\frac 1{\sqrt{(1-x)}^3}$$
$$g=sin^{-1}x$$ $$g'=\frac{1}{\sqrt{1-x^2}}$$
so anyway adding together
we get
$$\frac 1{(\sqrt{(1-x)}^3}*sin^-x+\frac x{\sqrt{(1-x^2)}}*\frac{1}{\sqrt... | A faster way is logarithmic differentiation
$$y=\frac{x \sin ^{-1}(x)}{\sqrt{1-x^2}}\implies \log(y)=\log(x)+\log(\sin ^{-1}(x))-\frac 12 \log(1-x^2)$$
$$\frac{y'}y=\frac 1 x+\frac{1 } {\sin ^{-1}(x)\sqrt{1-x^2}}+\frac x{1-x^2}=\frac {x \sqrt{1-x^2} +\sin ^{-1}(x)}{x \left(1-x^2\right) \sin ^{-1}(x) }$$
$$y'=\frac{y'}y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4614715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Integrating $\frac{x^3}{\sqrt{x^2 + 4x + 6}}$ Question:
Evaluate $\displaystyle\int \frac{x^3}{\sqrt{x^2 + 4x + 6}}\ dx $.
My attempt:
$\begin{align} \int \frac{x^3}{\sqrt{x^2 + 4x + 6}}\ dx & = \int \frac{x^3}{\sqrt{(x+2)^2 + 2}}\ dx \\& \overset{(1)}= \int \frac{(\sqrt{2} \tan(t) - 2 )^3 \sqrt{2} \sec^2(t)\ }{\sqrt{2... | By hyperbolic substitution
Letting $x+2=\sqrt 2 \sinh \theta$ transforms the integral into
$$
\begin{aligned}
I= & \int \frac{(\sqrt{2} \sinh \theta-2)^3}{\sqrt{2} \cosh \theta} \cdot \sqrt{2} \cosh \theta d \theta \\
= & \int\left(2 \sqrt{2} \sinh ^3 \theta-12 \sinh ^2 \theta+12 \sqrt{2} \sinh \theta-8\right) d \theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4615616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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find all triplets of positive rationals satisfying this property Find all $(x, y, z) \in Q^{+} | (a, b, c) \in Z^{+} $ for $ a = x + y + z, b = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}, c = xyz$.
I have tried using Viete's formulas, setting $x, y, z$ as the roots of some polynomial $p(t) = a_3t^3 + a_2t^2 + a_1t^2 + a_0... | You've made good progress. To finish, first use Vieta's formulas or just expand to get
$$\begin{equation}\begin{aligned}
p(t) & = (t-x)(t-y)(t-z) \\
& = t^3 - (x+y+z)t^2 + (yz+xz+xy)t - xyz
\end{aligned}\end{equation}\tag{1}\label{eq1A}$$
Note $bc = yz+xz+xy$ is an integer. Thus, all of the coefficients in \eqref{eq1A}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4617474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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The critical point $(0,0)$ is assymptotically stable. Demostrate $a_{11}+a_{22}<0$ and $a_{11}a_{22}-a_{12}a_{21}>0.$ I've got the following linear system:
$$\frac{dx}{dt}=a_{11}x+a_{12}y$$
$$\frac{dy}{dt}=a_{21}x+a_{22}y$$
The critical point $(0,0)$ is an assymptotically stable critical point of the system.
We have to... | Let's take an alternative approach without solving for the eigenvalues. The phrase "asymptotically stable" implies that Lyapunov's method can be used, in which the following conditions must be satisfied:
*
*$ V\left(\mathbf{0}\right) = 0 $
*$ V\left(\mathbf{x}\right) > 0, \quad \forall \; \mathbf{x} \neq \mathbf{0} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4617815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the intersection points of two circles Find the intersection points of the circles $$k_1:(x-4)^2+(y-1)^2=9\\k_2:(x-8)^2+(y+4)^2=100$$
The intersections point (if they exist) will satisfy the equations of both the circles, so we can find their coordinates by solving the system $$\begin{cases}(x-4)^2+(y-1)^2=9\\(x-... | Any point on the first circle : $P(4+3\cos t,1+3\sin t)$
If $P$ has to be on the second circle as well,
$$100=(4+3\cos t-8)^2+(1+3\sin t+4)^2=-24\cos t+30\sin t+16+25$$
$$\iff5\sin t-4\cos t=\dfrac{41}6$$
Now $|5\sin t-4\cos t|\le\sqrt{4^2+5^2}=\sqrt{41}$ which is $<\dfrac{41}6$ as $6<\sqrt{41}$
So, no real intersectio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4618115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
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Calculate the values of the serieses Given this function: $f(x)=|\sin x|$, I was supposed to find its cosinuses series, which came out $$\frac{2}{\pi}+\sum_{k=1}^\infty\frac{4}{\pi\left(1-4k^2\right)}\cos(2kx)\,.$$
After that, I was asked to find the values of these two serieses:
$$\sum_{n=1}^\infty\frac{1}{4n^2-1}$$
$... | Since $\;\displaystyle 0=f(0)=\frac{2}{\pi}+\sum_{k=1}^\infty\frac{4}{\pi\left(1-4k^2\right)}\,,\;$ it follows that
$\displaystyle\sum_{k=1}^\infty\frac{4}{\pi\left(4k^2-1\right)}=\frac2\pi\;\;,$
$\displaystyle\sum_{k=1}^\infty\frac1{4k^2-1}=\frac12\;.$
Hence,
$\displaystyle\sum_{n=1}^\infty\frac1{4n^2-1}=\frac12\;.$
M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4619555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Acute triangle $\triangle ABC$ has $(\sin A - 2 \sin 2B) = 2 - 2 \cos 2B$ Find the range of $\frac{\sin B + \sin C}{\sin A}$ Angles in acute triangle $\triangle ABC$ has $(\sin A - 2 \sin 2B) \tan A = 2 - 2 \cos 2B$
Find the range of $$\frac{\sin B + \sin C}{\sin A}$$
We can prove that $\frac{a^2}{bc} = 4$ through some... | Looks like you are very close.
You have established $\frac {\sin B + \sin C} {\sin A} = \frac {b+c}a$ using the Law of Sines.
From $\frac {a^2}{bc} = 4$ we obtain $a^2 = 4bc$ and $bc = \frac{a^2}4$.
Now:
\begin{align}
1 &> \cos A\\ &= \frac {b^2 + c^2 - a^2}{2bc}\\
&=\frac {b^2 + c^2 - 4bc}{2bc}\\
&=\frac {b^2 + 2bc + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4625234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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prove $(\sin x)^{-2}-x^{-2}\leq 1-\frac{4}{{\pi}^{2}},x\in(0,\pi/2]$ $(\sin x)^{-2}-x^{-2}\leq 1-\frac{4}{{\pi}^{2}},x\in(0,\pi/2]$
How to deal with this problem?
Observing that when $x=\pi/2$, the above inequality becomes equality.
Firstly, denote $f(x)=(\sin x)^{-2}-x^{-2}$ and then take derivative of $f(x)$. We have... | Let $f(x)=\sin^{-2}x-x^{-2}$.
Note
$$ f'(x)=\frac{2}{x^3}-\frac{2\cos x}{\sin^3x}=\frac{2(\sin^3x-x^3\cos x)}{x^3\sin^3x}=\frac{g(x)}{x^3\sin^3x}. $$
where
$$ g(x)=\sin^3x-x^3\cos x.$$
Now
$$ g'(x)=x^3\sin x-3\cos x(x^2-\sin^2x)\ge0, \forall x\in(0,\frac\pi2] $$
which implies that $g(x)$ is increasing and hence
$$ g(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4626032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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The projective plane as a smooth surface in a 4-dimensional space For $(a, b, c)$ pairwise distinct:
$$
\begin{align}
f: & S^2 & \mapsto & \quad \mathbb{R}^4 \\
& (x, y, z) & \mapsto & \quad (X, Y, Z, W) = (y \cdot z, x \cdot z, x \cdot y, a \cdot x^2 + b \cdot y^2 + c \cdot z^2)
\end{align}
$$
is a smooth dou... | Response to question 1
This is a complement to the excellent answer of Federico Fallucca.
The ideal characterizing the image of $f()$ is already generated by
the following four polynomials:
$$
\begin{align}
& (b - c)^2X^3 + (c - a)^2XY^2 + (a - b)^2XZ^2 + (W - b)(W - c)X - (a - b)(a - c)YZ & (C_0) \\
& (c - a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4627844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Prove $\frac{ab^2+2}{a+c} +\frac{bc^2+2}{b+a} +\frac{ca^2+2}{c+b} \geq \frac{9}{2}$ for $a,b,c\geq1$ Prove $\dfrac{ab^2+2}{a+c} +\dfrac{bc^2+2}{b+a} +\dfrac{ca^2+2}{c+b} \geq \dfrac{9}{2}$ for $a,b,c\geq1$.
I tried using the Titu Andreescu form of the Cauchy Schwarz inequality and got to this point:
$\dfrac{(b\sqrt{a}+... | Update: Add proof 2 without using Holder inequality.
Proof 2:
We have
\begin{align*}
&\frac{ab^2+2}{a+c} +\frac{bc^2+2}{b+a} +\frac{ca^2+2}{c+b}\\[6pt]
={}& \frac{ab^2}{a+c} +\frac{bc^2}{b+a} +\frac{ca^2}{c+b} + \frac{2}{a+c} + \frac{2}{b+a} + \frac{2}{c+b}\\[6pt]
\ge{}&3\sqrt[3]{\frac{ab^2}{a+c} \cdot\frac{bc^2}{b+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4628757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Sine curve that passes through (1,1) (2,2) and (3,3) Question: Find a sine function in the form $a\sin(bx) + c$ that passes through points (1,1) (2,2) and (3,3)
Working so far:
*
*We have three points for three unknown variables in the function, so we can use simultaneous equations to solve for them.
*Simultaneous ... | A little geometric intuition is helpful here. Your desired points are odd-symmetric around $(2,2)$, so you can imagine shifting your function up 2 steps and then scaling it horizontally to have period $4$ (instead of $2\pi$).
It would be straightforward then to shift the origin to $(2,2)$ and scale $x$ by $\pi/2$, but... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4630346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For what values of h are the three vectors linearly dependent?
For the vectors below find the value(s) of h for which the vectors are linearly dependent.
$$
\vec v_1=\left[\begin{array}{c} 2 \\ -4 \\ 1 \end{array} \right]\ \vec v_2 = \left[\begin{array}{c} -6 \\ 7 \\ -3 \end{array} \right]\ \vec v_3 = \left[\begin{arr... | Your attempt it is correct, we only need to interpret in terms of the definitions what is happening. We always can use the definition: we says that $\{v_1,v_2,v_3\}$ is linearly dependent if there exists scalars $\alpha_1,\alpha_2,\alpha_3$ not all zeros such that $\alpha_{1}v_1+\alpha_{2}v_2+\alpha_{3}v_3=0$. In ter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4631160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Calculate the following integral $ \int\frac{2x+1}{x^{n+2}(x+1)^{n+2}}\ln\left(\frac{2x^2+2x+1}{x^2(x+1)^2}+\frac7{16}\right)dx$ Hello I am trying to solve a pretty complicated integral. It is a from a set of problems, published in a monthly journal for high school students and they are exercises in preparation for a c... | As said in comments, using $u=x(x+1)$ as @David Quinn suggested the problem reduces to
$$I=\int\frac 1 {u^{n+2}} \log \left(\frac{7 u^2+32 u+16}{16 u^2}\right)\,du$$ which is
$$I=-\frac{2+(n+1) \log \left(16 u^2\right)}{(n+1)^2\, u^{n+1}}+\int \frac 1 {u^{n+2}}\log(7 u^2+32 u+16)\,du $$
$$J=\int \frac 1 {u^{n+2}}\log(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4631821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Equation of parabola passes through $4$ distinct points
Equation of axis of parabola which
passes through the point $(0,1)\ , \ (0,2)$
And $(2,0)\ ,\ (2,2)$ is
Let general equation of conic is
$ax^2+2hxy+by^2+2gx+2fy+c=0\cdots (1)$
And it represent parabola if $h^2=ab$
Parabola passes through $(0,1)$
Then put int... | I'm going to show a solution which uses the following claim, and then add a proof of the claim and an important fact which might interest you.
Claim : The equation of a parabola can be written as
$$\bigg(f(x,y)\bigg)^2+g(x,y)=0$$
where
$f(x,y)=0$ is the equation of the axis of symmetry, and $g(x,y)=0$ is the equation o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4635503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Is there any other simple method to evaluate $\int_0^{1} \frac{\ln ^n x}{\left(1-x^2\right)^2}dx $? After reading the fantastic solutions to the integral in the post,
$$\int_0^1 \frac{\ln x}{1 - x^2} \mathrm{d}x=-\frac{\pi^2}8,$$
and found that $$
\int_0^1 \frac{\ln ^2 x}{1-x^2} d x =\frac{7\zeta(3)}{4}. $$
Then I kee... | Utilize
\begin{align}
J(m)=\int_0^1 \frac{\ln^m x}{1-x^2}dx
=&\int_0^1 \frac{\ln^m x }{1-x}dx -\int_0^1 {\frac{x\ln^m x }{1-x^2} } \overset{x^2\to x}{dx} \\=& \ \left(1-\frac1{2^{m+1}}\right)\int_0^1\frac{\ln^mx}{1-x}dx\\
=&\ \left(1-\frac1{2^{m+1}}\right)(-1)^m m!\ \zeta(m+1)
\end{align}
to evaluate
\begin{align}
\... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 0
} |
Evaluating $\int_{-\infty}^{\infty}\frac{\ln\left(\frac{1}{2}+x+x^{2}\right)}{1+x^{2}}dx$ (Motivation) This is an integral I made up for fun. WolframAlpha doesn't seem to come up with a closed form for it and I'm surprised there doesn't seem to be a duplicate after using Approach0, but I believe it equals
$$\pi\ln\le... | There is even an antiderivative.
Write
$$\frac{\log \left(x^2+x+\frac{1}{2}\right)}{x^2+1}=\frac{\log \left((x-a)(x-b)\right)}{(x+i)(x-i)}$$ wtih
$$a=-\frac{1+i}{2} \qquad \text{and} \qquad b=-\frac{1-i}{2}$$
$$\frac 1{(x+i)(x-i)}=\frac i 2\left(\frac{1}{x+i}-\frac{1}{x-i}\right)$$ and face four integrals looking like
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4642590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Investigate on the convergence of $I(a)= \int_0^{\infty} \frac{x}{\sqrt{1+x^a}} d x $ for $a>0$ and its exact value in case of convergence. Inspired by the post,
I start to investigate the convergence of $$I(a)= \int_0^{\infty} \frac{x}{\sqrt{1+x^a}} d x $$ for $a>0$.
For any $0\le a\le 4$, if $x\ge 1$, we have
$$
\f... | The antiderivative
$$\int \frac{x}{\sqrt{1+x^a}} dx=\frac{1}{2} x^2 \,
_2F_1\left(\frac{1}{2},\frac{2}{a};\frac{a+2}{a};-x
^a\right)$$ leads to the same result for the definite integral.
What is interesting to notice is that, if $a=4+\epsilon$, the definite integral can be approximated as
$$\int_0^\infty \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4647411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why is the volume of a sphere $\frac{4}{3}\pi r^3$? I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for the formula?
| Let us consider the following sphere of radius r with an inscribed pyramid with base area G:
We can calculate the volume of the inscribed pyramid as follows:
$V_{pyramid} = \frac{1}{3} \cdot G \cdot r$
Notice that we can now express $V_{sphere}$ as the sum of the volume of infinitely small pyramids over the total surf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "108",
"answer_count": 19,
"answer_id": 18
} |
Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $ I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges,... | I want to use the infinite series expansion of the integral of a function to compute the sum. If you repeat the process of integration by parts over and over again:
$$\int f(x) dx = xf(x)-\int xf'(x)dx = xf(x) - \frac{x^2}{2}f'(x)+\int \frac{x^2}{2}f''(x)dx$$
$$\Rightarrow \int f(x) dx = xf(x) - \frac{x^2}{2}f'(x) + \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "60",
"answer_count": 12,
"answer_id": 8
} |
Summing the series $ \frac{1}{2n+1} + \frac{1}{2} \cdot \frac{1}{2n+3} + \cdots \ \text{ad inf}$ How does one sum the given series: $$ \frac{1}{2n+1} + \frac{1}{2} \cdot \frac{1}{2n+3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{1}{2n+5} + \frac{ 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{1}{2n+7} + \cdots \ \text{ad inf}$$
G... | Observe that
$$\displaystyle \frac{1}{4^n} {2n \choose n} = \frac{(2n-1)(2n-3)(2n-5)...}{2n(2n-2)(2n-4)...}$$
and recall that
$$\displaystyle \frac{1}{ \sqrt{1 - x^2} } = \sum_{k \ge 0} \frac{1}{4^k} {2k \choose k} x^{2k}.$$
It follows that the desired quantity is
$$\displaystyle \int_0^1 \frac{x^{2n}}{\sqrt{1 - x^2}} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
How do you show that $d\theta = \frac{x dy - y dx }{x^2 + y^2}$? If $(r, \theta)$ are polar coordinates on $\mathbb{R}^2\setminus \{ (0,0)\}$, then how do I show/prove that
\begin{equation*}
d\theta =\dfrac{x dy - y dx}{x^2 + y^2}?
\end{equation*}
| On $R^2\setminus\{0\}$ we have $\theta=\Im(\ln(x+iy))$ (here $\Im$ stands for "the imaginary part of"). Therefore $$d\theta = \Im\left(\frac1{x+iy}\right)dx+\Im\left(\frac{i}{x+iy}\right)dy=\Im\left(\frac{x-iy}{x^2+y^2}\right)dx+\Im\left(\frac{i(x-iy)}{x^2+y^2}\right)dy=$$$$\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/6083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $. How does one prove the following limit?
$$
\lim_{n \to \infty}
\sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}}
= 3.
$$
| Let me provide a full and simple proof here (6 years later)
Set, for $m<n$
$$
a_{m,n}=\sqrt{1+m\sqrt{1+(m+1)\sqrt{1+\cdots+(n-1)\sqrt{1+n}}}}.
$$
We shall first show that $$a_{m,n}<m+1,\tag{1}$$ in the following way using
Backwards induction. Clearly,
$a_{n,n}=\sqrt{1+n}<1+n$, and if $a_{k+1,n}<k+2$, for some $k<n$, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/7204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "54",
"answer_count": 3,
"answer_id": 2
} |
Does Stirling's formula give the correct number of digits for $n!\phantom{}$? It is known that the number of digits of a natural number $n > 0$, which represent by $d(n)$ is given by:
$d(n)= 1 + \lfloor\log n\rfloor\qquad (\text{I})$
($\log$ indicates $\log$ base $10$)
Well .. the classical approach to the Stirling f... | Here are some thoughts on the conjecture that may lead one to suspect that it is true. This is not a proof that it is true.
We want to know whether
$$ \left\lfloor \log_{10} n! \right\rfloor = \left\lfloor \log_{10} \lfloor f(n) \rfloor \right\rfloor$$
is true for all $n > 1.$
We note that this would NOT be true if the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 0
} |
Evaluate $\int \frac{1}{\sin x\cos x} dx $ Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx $
I know that the correct answer can be obtained by doing:
$\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However... | Another way to integrate would be to let $\displaystyle t = \frac{\sin{x}}{\cos{x}}$, then $\displaystyle \frac{dt}{dx} = \frac{1}{\cos^2{x}} \Rightarrow dx = \cos^2{x}\;{dt}$.
Thus $ \displaystyle I = \int\frac{\cos^2{x}}{\sin{x}}\;{dt} = \int\frac{\cos{x}}{\sin{x}}\;{dt} = \int\frac{1}{t}\;{dt} = \ln{t}+k = \ln\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 10,
"answer_id": 6
} |
$n^2 + 3n +5$ is not divisible by $121$ Question:
Show that $n^2 + 3n + 5$ is not divisible by $121$, where $n$ is an integer.
| Make a contradiction that $n^2 + 3n + 5$ is divisible by $121$
Let $k$ be any positive integer, we can say that
$n^2 + 3n + 5 = 121\cdot k$
$n^2 + 3n + (5 - (121\cdot k)) = 0$
Solve for $n$,
$$\begin{align}
n=&\frac{-3 \pm \sqrt {(3)^2 - 4\cdot1\cdot(5-(121\cdot k))}}{2\cdot1}\\
n=&\frac{-3 \pm \sqrt {(484\cdot k)-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
$x^y = y^x$ for integers $x$ and $y$ We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
| Although this thing has already been answered, here a shorter proof
Because $x^y = y^x $ is symmetric we first demand that $x>y$
Then we proceed simply this way:
$ x^y = y^x $
$ x = y^{\frac x y } $
$ \frac x y = y^{\frac x y -1} $
$ \frac x y -1 = y^{\frac x y -1} - 1 $
Now we expand t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "92",
"answer_count": 6,
"answer_id": 3
} |
Summing up the series $a_{3k}$ where $\log(1-x+x^2) = \sum a_k x^k$ If $\ln(1-x+x^2) = a_1x+a_2x^2 + \cdots \text{ then } a_3+a_6+a_9+a_{12} + \cdots = $ ?
My approach is to write $1-x+x^2 = \frac{1+x^3}{1+x}$ then expanding the respective logarithms,I got a series (of coefficient) which is nothing but $\frac{2}{3}\ln... | Lemma: Let $f(x) = \sum a_n x^n$ be a power series. Then
$$\sum a_{3n} x^{3n} = \frac{f(x) + f(\omega x) + f(\omega^2 x)}{3}$$
where $\omega = e^{ \frac{2\pi i}{3} }$.
Proof. Ignoring convergence it suffices to prove this for a single term, and then it boils down to the identity
$$\frac{1 + \omega^n + \omega^{2n}}{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/11739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Is this definite integral really independent of a parameter? How can it be shown? I want to find a nice simple expression for the definite integral
$$\int_0^\infty \frac{x^2\,dx}{(x^2-a^2)^2 + x^2}$$
Now, I can numerically compute this integral, and it seems to converge to $\pi/2$ for all real values of $a$. Is this i... | Yes it is true!
Let
$$\displaystyle I = \int_{0}^{\infty} \dfrac{x^2}{x^2 + (x^2-a^2)^2} \ \text{dx}$$
Make the substitution $\displaystyle x = \dfrac{a^2}{t}$
We get
$$\displaystyle I = \int_{0}^{\infty} \dfrac{a^6}{t^4\left(\dfrac{a^4}{t^2} + \left(\dfrac{a^4}{t^2} - a^2\right)^2\right)} \ \text{dt} = \int_{0}^{\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/13414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 3,
"answer_id": 1
} |
Find the vertical asymptote of a function For an assignment, I was asked to find the vertical asymptote of the function $$g(x)= \frac{\frac{1}{2}x^3-4x^2+6x}{7x^2-56x+84}.$$
According to my text, a reliable method of finding the asymptote is to factor the numerator and denominator, and what left in the denominator tha... | You did not finish factoring the numerator. That's where the problem is.
You started out all right, but then you had to keep factoring that $x^2-6x$: it is not degree $1$, and it is not irreducible quadratic, so it can still be factored. In fact, it has a factor of $x$. So you really have:
$$\frac{1}{2}x^3 - 4x^2+6x = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How to compute the following definite integral Studying some integral table, I came across the following definite integral
$$\int_0^{\pi} \log [ a^2 + b^2 -2 a b \cos \phi ]\,d\phi$$ for $a,b \in \mathbb{R}$. Does somebody know a nice way to get the results?
| Here is a different way, and is strangely the first thing that came to my mind. It uses generating series, and power series as well as other various techniques:
(Warning!: It is significantly more complicated)
Assume that $a\geq b\geq0$ without loss of generality. Notice that our integral is $$\pi \log(a^2+b^2)+\int_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/23599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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$n^3 \equiv n^5 \pmod{12} $? I am proving that
$$5n^3 + 7n^5 \equiv 0 \pmod{12}$$
It would suffice to show
$$n^3 \equiv n^5 \pmod{12}$$
How would I go about doing that?
I suppose I could just go through each $n \equiv r \pmod{12}$ with $r$ from $1$ to $11$ and show that $n^3 \equiv n^5 \pmod{12}$ for each, but that w... | *
*$n^2 = 0,1,4$ or $9 \mod 12$ by checking each number $0,1,2,3,4,5,6$ (we don't have to check $7,8,9,10$ and $11$ since they are negatives and $x^2 = (-x)^2).$
*$n^4 = (n^2)^2 = n^2 \mod 12$ by checking each number 0,1,4 and 9.
*thus $n^3 = n^5.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/24100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
How do I come up with a function to count a pyramid of apples? My algebra book has a quick practical example at the beginning of the chapter on polynomials and their functions. Unfortunately it just says "this is why polynomial functions are important" and moves on. I'd like to know how to come up with the function (th... | The first layer has $1^2$ elements; the second has $2^2$ elements; the next has $3^2$ elements, and so on.
So this is a matter of finding the formula for
$$1^2 + 2^2 + 3^2 + \cdots + n^2$$
in terms of $n$.
For squares, this is a well known formula: $1^2+2^2+\cdots +n^2 = \frac{n(n+1)(2n+1)}{6}$.
Note that
$$\frac{n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/24521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 8,
"answer_id": 1
} |
Why does the sum of a division applied to individual items not equal the division applied to the sum of those items? When $a_2/a_1 = b_2/b_1$, $a_1 \neq b_1$, we have
$$\frac{a_{1}}{a_{2}/a_{1}}+\dfrac{b_{1}}{b_{2}/b_{1}}=
\frac{a_{1}+b_{1}}{1+\dfrac{(a_{2}+b_{2})-(a_{1}+b_{1})}{a_{1}+b_{1}}}.$$
So why when $a_2/a_1 \... | I'm not sure where the figures are coming from, but an interpretation is the inequality ($a,b,c > 0$)
$$\frac{a}{a+b} + \frac{a}{a+c} \ge \frac{2a}{a+ (b+c)/2}, \quad (1)$$
which follows from
$$((a+b)-(a+c))^2 \ge 0$$
and so
$$(a+b)^2+(a+c)^2 \ge 2(a+b)(b+c).$$
Therefore, on adding $2(a+b)(b+c)$ to both sides,
$$((a+b)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/24789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Proving $(2n+1) + (2n+3) + \cdots + (4n-1) = 3n^{2}$ for all positive integers $n$
Prove $(2n+1) + (2n+3) + (2n+5) +
\cdots + (4n-1) = 3n^{2}$ for all
positive integers $n$.
So the provided solution avoids induction and makes use of the fact that $1 + 3 + 5 + \cdots + (2n-1) = n^{2}$ however I cannot understand th... | The simple solution using $1+3+\ldots+(2n-1)=n^2$ is as follows:
$(2n+1)+(2n+3)+\ldots+(4n-1)=2n\cdot n+ (1+3+\ldots+(2n-1))=3n^2$
The fact that $1+3+\ldots+(2n-1)=n^2$ is true is a simple induction: Assume it's true for $n$. Then for $n+1$ we get $1+3+\ldots+(2n-1)+(2(n+1)-1)=n^2+2n+1=(n+1)^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/25623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 4
} |
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? How can I evaluate
$$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$?
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that ther... | If you want a solution that doesn't require derivatives or integrals, notice that
\begin{eqnarray}
1+2x+3x^2+4x^3+\dots = 1 + x + x^2 + x^3 + \dots \\
+ x + x^2+ x^3 + \dots\\
+ x^2 + x^3 + \dots \\
+x^3 + \dots \\
+ \dots \\
=1 + x + x^2 + x^3+\dots \\
+x(1+x+x^2+\dots) \\
+x^2(1+x+\dots)\\
+x^3(1+\dots)\\
+\dots \\
=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "438",
"answer_count": 23,
"answer_id": 14
} |
Finding the cos angle between two matrices using the euclidean inner product I wanted to know if I did this problem right or not.
$A$ and $B$ are the following matrices:
$A=\begin{bmatrix}2&6\\1&-3\end{bmatrix},$
$B=\begin{bmatrix}3&2\\1&0\end{bmatrix}.$
Then
$$\frac{\langle A,B\rangle}{\|A\|\|B\|}
=\frac{\left\lang... | $a\cdot b=|a||b|\cos\theta$ so
$$\theta=\arccos\Big(\frac{a\cdot b}{|a||b|}\Big).$$
we have
$a\cdot b=6+12+1=19, |a||b|=\sqrt{4+36+1+9}\sqrt{9+4+1}=10\sqrt{7}$.
so the angle is $\arccos(19/(10\sqrt{7}))$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/31001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Manifold with minimum surface distance between two points The book "The World is Flat" uses flatness as a metaphor for a global economy. In fact, a spherical world would seem to be better than a flat world in terms of reducing the distances between two random points on the surface of the world. The shorter the distance... | The average distance on a sphere of radius $r$ is $\frac{\pi}{2}r$ (which we can get without any integrations because there's as much area at a distance $\frac{\pi}{2}+\theta$ from a given point as there is at $\frac{\pi}{2}-\theta$); so this is
$$\bar{d}=\frac{\pi}{2}\sqrt{\frac{A}{4\pi}}=\frac{\sqrt{\pi}}{4}\sqrt{A}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/36708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
How to prove $\left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor$? everybody, how can I prove that, for natural $m$ and $n$,
$$
\left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor \; ?
$$
Thanks a lot.
| what about Mathematical induction prove :
n and m is natural number :
\begin{aligned} m=1 \rightarrow \left \lfloor \frac{1+n-1}{1} \right \rfloor = \left \lceil \frac{n}{1} \right \rceil \rightarrow true \end{aligned}
\begin{aligned}m=2 \rightarrow \left \lfloor \frac{2+n-1}{2} \right \rfloor = \left \lceil \frac{n}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/37555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 9,
"answer_id": 6
} |
Bell Numbers and Equating Coefficients I need help solving the following exercise:
Let B(n) be the Bell - Numbers of $[n]$ and $$\exp(x) := \sum_{n\geq 0} \frac{x^n}{n!}$$ the exponential function.
Prove by equating the coefficients and using
$$ B(n) = \sum\limits_{a_1+\cdots+a_k=n, a_i \geq 1} \frac{1}{k!} \binom{n}{a... | We are interested in expansion of the term $\exp(\exp(x)-1)$, now $$\exp(x) - 1 = \sum_{n=1}^\infty \frac{x^n}{n!}$$ and since $\displaystyle \exp(z) = \sum_{m=0}^\infty \frac{z^m}{m!}$ we should start by considering $\displaystyle \left(\sum_{n=1}^\infty \frac{x^n}{n!}\right)^m$ for $m=2,3,\ldots$, here is a table jus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/39250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Lower bound over a convex function I will be to grateful if help me find a tight lower bound $g(x)$ over the following convex function:
$$f(x) = \sqrt{1+4x^2} -1 + \log(\sqrt{1+4x^2}-1) - \log(2x^2) \geq g(x),$$
where $g(x)$ is preferably a polynomial with degree of at least $2$.
The taylor expansion around the point... | For the case $|x|\leq 1$ I propose the following: Write $g(x):= a x^2- b x^4$ and choose $a$ and $b$ such that $f(1)=g(1)$, $f'(1)=g'(1)$. I did this using Mathematica and obtained
$$a=2\Bigl({3\over 1+\sqrt{5}} +\log {2\over 1+\sqrt{5}}\Bigr)\ \qquad b=
{2\over 1+\sqrt{5}} +\log{2\over 1+\sqrt{5}}\ .$$
Plotting the ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can we make an integral domain with any number of members? Is it true that rings without zero divisors (integral domains) can have any number of members except for 4,6? and if this is true then what would the multiplication operator be?
| Another way to define field F4 is by defining addition with:
$$ 0 + a = a $$
$$ a + a = 0 $$
$$ 1 + 2 = 3 $$
and multiplication having:
$$ 0 \times a = 0 $$
$$ 1 \times a = a $$
$$ 2 \times 2 = 3 $$
$$
\begin{array}{c|cccc}
{\Large +} & \overline{0} & \overline{1} & \overline{2} & \overline{3} \... | {
"language": "en",
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Continued proportion implies $(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2$ I am trying to find a tricky way to proof these:
If $a,b,c,d$ are in continued proportion, prove that $$(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2 ,$$ This result could be extended to $$(a^2+b^2+c^2+d^2)(b^2+c^2+d^2+e^2)=(ab+bc+cd+de)^2$$ when $a,b,c... | You know that $b=ka$, $c=kb$ etc so the lhs can be rewritten
$$(a^2+b^2+c^2)(k^2a^2+k^2b^2+k^2c^2) = k^2(a^2+b^2+c^2)^2$$
and the rhs can be written
$$(ka^2 + kb^2 + kc^2)^2 = k^2(a^2+b^2+c^2)^2$$
and you're done. The same trick works for the second example.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do I get the square root of a complex number? If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
| Write $(a+bi)^2 = 9+4i$. Then $a^2-b^2 +2abi = 9+4i$.
By comparison of coefficients,
$2ab = 4$ and $a^2-b^2 = 9$.
Thus $ab=2$ and $a^2 = 9 +b^2$.
Setting $a = 2/b$ with $b\ne 0$ gives $4/b^2 = 9 + b^2$ and so $4 = 9b^2 + b^4$, i.e., $b^4 + 9b^2 -4 = 0$.
Solve $x^2 + 9x-4=0$, where $x=b^2$. Solutions are
$x_{1,2} = \fr... | {
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"source": "stackexchange",
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"answer_id": 11
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Basic question about mod I'm having a tough time understanding why $(a^x \bmod p)^y \bmod p$ is equal to $a^{xy}\bmod p$.
Does this have a mathematical proof? Please advise.
| It's usually simpler to work with congruences rather than the binary $\bmod$ operator, and then go tot he binary operator at the end.
Given an integer $n$, we say that $a\equiv b\pmod{n}$ ("$a$ is congruent to $b$ modulo $n$") if and only if $n$ divides $b-a$. It is easy to show that "congruent modulo $n$" is an equiva... | {
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"timestamp": "2023-03-29T00:00:00",
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Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$ I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:
$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
I really have no idea why this statement is true. Can someone please explain ... | Late to the party, but I hope I am contributing something new to the answers already posted here.
We seek
$$\sum_{i=1}^Ni^2=1^2+2^2+\cdots+N^2$$
which, geometrically speaking, is equivalent to adding the area of $N$ squares with side lengths $1,2,...,N$. A visual depiction when $N=2$ is
... | {
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"source": "stackexchange",
"question_score": "145",
"answer_count": 32,
"answer_id": 14
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Expansion concerning the binomial theorem The question goes:
Expand $(1-2x)^{1/2}-(1-3x)^{2/3}$ as far as the 4th term.
Ans: $x + x^2/2 + 5x^3/6 + 41x^4/24$
How should I do it?
| Note that:
*
*$\displaystyle (1-x)^{\frac{p}{q}} = 1 - \frac{p}{q} \cdot \frac{x}{1!} + \frac{ \frac{p}{q} \cdot \Bigl(\frac{p}{q} -1\Bigr)}{2!} \cdot x^{2} - \cdots $
Using this formula you have $$(1-2x)^{1/2} = 1- \frac{1}{2} \cdot \frac{2x}{1!} + \frac{\frac{1}{2} \cdot \Bigl(-\frac{1}{2}\Bigr)}{2!}\cdot... | {
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Proof by induction that $2^n \gt n^k$ How do I prove using induction that if $k$ is a natural number, then $2^n \gt n^k$ for all $n \geq k^2 + 1$, where $n$ is also a natural number?
| If found only this proof, which is rather cumbersome. I hoppe that I did not make mistake there and that someone will come up with a more elegant solution.
Lemma 1: $2^k\ge k^2$ for any $k\ge 4$.
EDIT: In the comments you can find a nice combinatorial argument provided by Aryabhata which works for $k\ge5$.
Proof by in... | {
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General approach for problem for finding sum from $1$ to $N$ when all $a$'s are replaced by $b$'s The problem is:
Find the sum of all the numbers from $1$ to $100$ when all the $6$'s are replaced by $9$'s.
I need some ideas on how to approach this kind of problems? Please explain your ideas, assuming one digit replac... | Hint:
*
*Can you find the sum $1 + 2 + \dots 100$?
*What are all the values between $1$ and $100$ that have a $6$ in them? List them.
(Edit:) The process is the same for the numbers $1, \dots , 10$. You would add up all the values $1 + \dots + 10 = \frac{10 \cdot 11}{2} = 55$. Then, subtract each number with a ... | {
"language": "en",
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$x^4+4$ is composite for $x>1$
$x^4+4$ is composite for $x>1$
I know the Sophie Germain indentity and the get the factorization $$x^4+4 = (x^2+2-2x)(x^2+2+2x)$$
But I am stuck here. I cannot see any general factor here.
| A version in two variables is
$$ ( x^2 + 2 x y + 2 y^2) (x^2 - 2 x y + 2 y^2) = x^4 + 4 y^4 $$
where both quadratic forms are positive definite, so that there are only a finite number of $(x,y)$ pairs such that either factor is $1.$
| {
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"source": "stackexchange",
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A "fast" way to ,find the maximum value of $(x^2) \times (y^3)$, if $3x+4y=12$ for $x,y \ge 0$ If $3x+4y=12$ $\forall x,y \ge 0$, the maximum value of $(x^2) \times (y^3)$ is
*
*$6 \times (6/5)^5$
*$3 \times (6/5)^5$
*$ (6/5)^5 $
*$7 \times (6/5)^5$
How to approach this problem? I thought of using the approac... | AM-GM gives
$$12 = \frac{3}{2} x + \frac{3}{2} x + \frac{4}{3} y + \frac{4}{3} y + \frac{4}{3} y \ge 5 \left( \frac{16}{3} x^2 y^3 \right)^{1/5}$$
hence
$$x^2 y^3 \le \frac{3}{16} \left( \frac{12}{5} \right)^5 = 6 \cdot \left( \frac{6}{5} \right)^5$$
with equality when $\frac{3}{2} x = \frac{4}{3} y$, which is attaina... | {
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Challenging inequality: $abcde=1$, show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{33}{2(a+b+c+d+e)}\ge{\frac{{83}}{10}}$ Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that:
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}... | This is only a partial solution but I think someone more familiar with such elementary inequalities than myself might be able to finish it. You can replace $a,b,c,d$, and $e$ with their reciprocals and the inequality in question becomes
$$a + b + c + d + e + {33 \over 2}{1 \over ({1 \over a} + {1 \over b} + {1 \over c}... | {
"language": "en",
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In a acute angled triangle, we have $\tan(A)\cdot\tan(B)\cdot\tan(C) \geq 3\sqrt{3}$ How to show:
In a acute angled $\triangle \ ABC$ show that $$\tan(A) \cdot \tan(B)\cdot \tan(C) \geq 3\sqrt{3}$$
Any ideas?
| $$A+B=\pi-C$$
\begin{align*}
&\tan (A+B)= \tan (\pi-C)\\
&(\tan A+ \tan B)/(1-\tan A \tan B)= (\tan \pi- \tan C)/(1+\tan \pi \tan C)=-\tan C\\
&(\tan A+ \tan B)= -\tan C(1-\tan A \tan B)\\
&\tan A + \tan B= -\tan C+ \tan A \tan B \tan C\\
&\tan A + \tan B+ \tan C= \tan A \tan B \tan C
\end{al... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Limit of $(x+3)^{1 + 1/x} - x^{1 + 1/(x+3)}$ when $x\to \infty$ Okay, this is the last limit I have to solve but it's not that easy ;)
$$\lim_{x\to \infty} (x+3)^{1 + 1/x} - x^{1 + 1/(x+3)}$$
| For constants a, b, consider the asymptotics of $(x+a)^{1+1/(x+b)}$ as $x\to\infty$. This is equal to $(x+a)e^{\log(x+a)/(x+b)}$.
Now since $\log(x+a)/(x+b)$ tends to 0, we know that $e^{\log(x+a)/(x+b)}=1+\log(x+a)/(x+b)+\ldots$ where the $\ldots$ term tends to 0 faster than $\log(x+a)^2/(x+b)^2$.
Then $(x+a)^{1+1/(x... | {
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Finding limit of a quotient I can't find this for some reason. I know I asked about 6 of these before, and I was able to finish my homework but now I went back to review and I can't do a single one of these problems on my own. Even the ones I did figure out on my own. I spent probably a total of 14 hours on the homewor... | From the comments, it appears that it would be useful to give a fairly elementary and pedantic discussion of using the factor theorem (from precalculus or college algebra, in the U.S.).
Factor Theorem: If $r$ is a zero of a polynomial $P(x)$ (i.e. a solution to $P(x) = 0$ is $x=r$), then $x-r$ is a factor of $P(x)$.
Ex... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Fourier transform of a wavefunction It would be great if someone would help me check that I have done this right.
I have an infinite square well, s.t.
$V(x) = 0$ for $x\in(0,a)$ and $V(x) = \infty$ otherwise.
Given that $\psi(x,0) = \alpha x (a-x)$, where $\alpha\in\mathbb R$
I have found by normalization that $\alpha... | I think you messed up on the normalization. I think it's $\alpha=6/a^3$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
| Hint: $(1+2+...+n+(n+1))^2 - (1+2+...+n)^2 = 2(1+2+...+n)(n+1) + (n+1)^2 = (n+1)(2(1+2+...+n) + (n+1)) = (n+1)(n(n+1) + n+1) = (n+1)(n+1)^2 = (n+1)^3$
We use here the (slightly easier to prove) result $\sum_{k=1}^{n} k = n(n+1)/2$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "67",
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"answer_id": 10
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Finding limits of rational functions as $x\to\infty$
Possible Duplicate:
Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials
$$\displaystyle \lim_{x\to\infty}\frac{(2x^2+1)^2}{(x-1)^2(x^2+x)}.$$
I do not know where to start on this, I tried multiplying it out and that didn't help really. It seems ... | By substitution.
I think the OP is having some difficulty in dividing by a power of $x$ and simplifying the rational function, in terms of $1/x$. I must admit that the simplification process seems a bit unnatural while working with $1/x$.
So, my suggestion is that, at the very beginning, substitute $y = 1/x$. Then $x ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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The square of an integer has form $3n$ or $3n+1$ Prove that if $n\in\mathbb Z$, then $n^2$ is of the form $3q$ or $3q+1$ for some $q\in\mathbb Z$
I would like to show that 3q+2 is = 3q+1 thus $n^2$ can be of the form of 3q or 3q+1.
Case one
$(3k)^2=(3k)(3k)=9k^2=3(3k^2)$ and is still of the form $3q$
When $q=3k^2$
$3q... | Proof is easy ($k \in Z$)
1) if $n = 3k$, $n^2 = 9k^2$, QED
2) if $n = 3k + 1$, then $n^2 = 3kn + n = 3kn + 3k + 1$, QED
3) if $n = 3k + 2$, then $n^2 = 3kn + 2n = 3kn + 6k + 3 + 1$, QED
| {
"language": "en",
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"source": "stackexchange",
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Simplify with no calculator $\dfrac{(8^3)(-16)^5}{4(-2)^8}$
$\dfrac{8\cdot8\cdot8\cdot-16\cdot-16\cdot-16\cdot-16\cdot-16}{4\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2}$
$\dfrac{8\cdot8\cdot8\cdot-16\cdot-16\cdot-16\cdot-16\cdot-16}{2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2}$
$\dfrac{8\cd... | When multiplying fractions, cancel BEFORE multiplying.
$$
\dfrac{8}{2} \cdot \dfrac{8}{2} \cdot \dfrac{8}{2} \cdot \dfrac{-16}{2} \cdot \dfrac{-16}{2} \cdot \dfrac{-16}{2} \cdot \dfrac{-16}{2} \cdot \dfrac{-16}{2}
$$
Wherever you see $\dfrac{8}{2}$, put $4$.
Wherever you see $\dfrac{-16}{2}$, put $-8$.
Then you have
... | {
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Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $ How could we prove that this inequality holds
$$ \left(1+\frac{1}{n+1}\right)^{n+1} \gt \left(1+\frac{1}{n} \right)^{n} $$
where $n \in \mathbb{N}$, I think we could use the AM-GM inequality for this but not getting how?
| This is equivalent to showing that
$$
\left(\frac{n}{n-1}\right)^{n-1}\tag{1}
$$
is an increasing function of $n$. Consider the Taylor expansion of
$$
\begin{align}
(n-1)\log\left(\frac{1}{1-1/n}\right)
&=(n-1)\left(\frac1n+\frac12\frac1{n^2}+\frac13\frac1{n^3}+\dots\right)\\
&=1-\frac1{1\cdot2}\frac1n-\frac1{2\cdot3}\... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 6,
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Proving a trigonometric identity How can one prove the validity of this trigonometric identity?
\begin{equation}
2\arccos\sqrt{x} = \frac{\pi }{2}-\arcsin(2x-1)
\end{equation}
| EDITED in response to valdo's answer.
Your identity $$
2\arccos \sqrt{x}=\frac{\pi }{2}-\arcsin (2x-1),\qquad 0\le x\le 1\tag{0},
$$ may be rewritten as $$
\arcsin (2x-1)=\frac{\pi }{2}-2\arccos \sqrt{x},\qquad 0\le x\le 1\tag{1}.
$$
For identity $(1)$ to be valid$^1$ it is enough that
$$
\sin \left( \arcsin ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 1
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Residue integral problem Have this Integral.
$$\int_{-\infty }^\infty \frac{\cos(2x)}{(x^2+1)(x^2+4)^2}\,dx$$
Been working on similar problems but the cos bother me in this problem. Can anyone help me get started? What should I do first?
I´m given that i could use
$$\operatorname{Re}{\left ( \int_{-\infty }^\infty \f... | Very much like the example II in this wiki page, we write:
$$
\int_C \frac{\mathrm{e}^{2 i z}}{(z^2+1)(z^2+4)^2} \mathrm{d} z =
\int_{-\infty}^\infty \frac{\mathrm{e}^{2 i x}}{(x^2+1)(x^2+4)^2} \mathrm{d} x + \int_{0}^{\pi} \frac{\exp(2 i R \mathrm{e}^{i \varphi} )}{((R \mathrm{e}^{i \varphi})^2+1)((R \mathrm{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How many ordered pairs of positive integers
For a prime integer p, how many ordered pairs of positive integers
(a, b) are there that satisfy $$\frac{1}{a} + \frac{1}{b} =\frac{1}{p}$$
For example, for p = 5, $$\frac{1}{6} + \frac{1}{30}$$ and
$$\frac{1}{30} + \frac{1}{6}$$ are two different ways of getting
$\fra... | If we multiply through by $pab$, we find that for $a, b \ne 0$, our equation is equivalent to $pb+pa=ab$, which we can rewrite as
$$(a-p)(b-p)=p^2.$$
We are looking for non-zero solutions of this equation. Conveniently, $p^2$ does not have many factorizations!
We can have $a-p=-1$, $b-p=-p^2$, which yields a negative ... | {
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"timestamp": "2023-03-29T00:00:00",
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A pencil approach to find $\sum \limits_{i=1}^{69} \sqrt{\left( 1+\frac{1}{i^2}+\frac{1}{(i+1)^2}\right)}$ What is the fastest, paper-pencil method of finding $$\sum \limits_{i=1}^{69} \sqrt{\left( 1+\frac{1}{i^2}+\frac{1}{(i+1)^2}\right)}?$$
This is actually a quantitative aptitude problem, and hence the solutions sho... | This is a late response (since I joined three weeks ago) but here is another method. Write
\begin{align}
1 + \frac{1}{i^2} + \frac{1}{(i + 1)^2} &= 1 + \frac{2}{i(i+1)} + \left(\frac{1}{i^2} - \frac{2}{i(i+1)} + \frac{1}{(i+1)^2}\right)\\
&= 1 + 2\left(\frac{1}{i} - \frac{1}{i+1}\right) + \left(\frac{1}{i} - \frac{1}{i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
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General formula to obtain triangular-square numbers I am trying to find a general formula for triangular square numbers. I have calculated some terms of the triangular-square sequence ($TS_n$):
$TS_n=$1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056
1882672131025, 63955431761796, 2172602007770041, 7380451... | $${t_n} = \sum\limits_{k = 1}^n k = 1 + 2 + 3 + \cdots + n = \frac{{n\left( {n + 1} \right)}}{2} = n-th{\text{ triangular number}}$$
$${s_m} = {\text{m-th square number}} = {m^2}$$
$${s_m} = {t_n} \Rightarrow \frac{1}{2}n\left( {n + 1} \right) = {m^2}$$
$$\frac{1}{2}{\left( {n + \frac{1}{2}} \right)^2} = \frac{1}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/76040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 3
} |
Integrate $\int\frac{1}{x^6} \sqrt{(1-x^2)^3} ~ dx$ How to integrate the following?
$$\int\frac{\sqrt{(1-x^2)^3}}{x^6} \;dx .$$
| Integral,
$$
\begin{align*}
I &= \int \frac{\sqrt{(1-x^2)^{3}}}{x^6} dx
\\ &= \int \left( \frac{1-x^2}{x^2} \right)^{3/2} \frac{1}{x^3}dx
\\ &= \int \left(\frac{1}{x^2}-1 \right)^{3/2} \cdot \frac{1}{x^3} dx
\end{align*}
$$
Make the substitution: $z= \frac{1}{x^2}-1$, so that $dz=\frac{-2}{x^3} ~dx$.
$$
\begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Solving $(2y-4)(2y+1) = (2y-2)^2$ I'm getting different answer from answer key.
Solving $$(2y-4)(2y+1) = (2y-2)^2$$
FOIL left side $$4y^2+2y-8y-4 = (2y-2)^2$$
Right side $$4y^2+2y-8y-4 = 4y^2+4 $$
Subtract $4y^2$ from both sides
$$2y-8y-4 = 4 $$
Combine $y$
$$6y-4 = 4$$
add 4 to both sides
$$6y = 8$$
But the answer key... | You're unfolding the right-hand side wrong -- $(2y-2)^2$ is not $4y^2+4$, but $4y^2+4-8y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/77570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
chances of getting three of one kind and four of another out of seven dice There are several questions similar to this one but after reading those,
I am still very confused.
I also did a similar problem of this one and I think I got it, but then I got stuck again.
So if four dice are rolled, the chance of getting thre... | There are $6^7$ possible outcomes, listed as 7-tuples. Now, how many 7-tuples would consist of two distinct elements, 4 of one kind, and 3 of other. Such 7-tuples can be ordered into $[a,a,a,a,b,b,b]$ with $a \not= b$. $a$ can be chosen 6 different ways. Once $a$ is chosen, $b$ can be chosen 5 different ways. There are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Determinant of a specific circulant matrix, $A_n$ Let
$$A_2 = \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]$$
$$A_3 = \left[ \begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$
$$A_4 = \left[ \begin{array}{cccc} 0 & 1 & 1 & 1\\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0\end{array... | The determinant of a general circulant matrix is given by
\begin{eqnarray*}
\left| \begin{array} {ccccc}
x_0 & x_{n-1} & \cdots & x_2 &x_1 \\
x_1 & x_0 &\cdots & x_3 & x_2\\
x_2 & x_1 & \cdots & x_4& x_3 \\
\vdots & \vdots & ~ & \vdots & \vdots \\
x_{n-2} & x_{n-3} &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 0
} |
Which is the "fastest" way to compute $\sum \limits_{i=1}^{10} \frac{10i-5}{2^{i+2}} $?
I am looking for the "fastest" paper-pencil approach to compute $$\sum \limits_{i=1}^{10} \frac{10i-5}{2^{i+2}} $$
This is a quantitative aptitude problem and the correct/required answer is $3.75$
In addition, I am also interested... | First note that $\displaystyle\frac{10i-5}{2^{i+2}} = \frac{5(2i-1)}{2^{i+2}}$
Secondly, using the sum of a geometric series you can show that $2^0 + 2^1 + 2^2 + ... + 2^k = 2^{k+1}-1$.
$\displaystyle \begin{align*} \text{So } \sum \limits_{i=1}^{n} \frac{10i-5}{2^{i+2}}
&= 5(\frac{2\times1-1}{2^{1+2}}+\frac{2\time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$ $5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$
Work out the values of $\frac{1}{x+y}$
| Initial Solution:
$5^x+2^y=7/10$
$5^x+2^y=(5+2)/(5*2)=(1/2+1/5)=5^{-1}+2^{-1}$ ------ (1)
$5^y+2^x=(5+2)/(5*2)=(1/5+1/2)=5^{-1}+2^{-1}$ ------ (2)
By examining (1) and (2) we get the solution:
$x=y=-1$
Now we consider the relation:
$5^x+2^y=k$ --------(1)
where k is a constant.
$\frac{dy}{dx}=-\frac{5^x}{2^y}log_{2} 5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 6,
"answer_id": 2
} |
Problem based on Range Find $a$ and $b$ such that the inequality $a \le 3 \cos{x} + 5\cos\left(x - \frac{\pi}{6}\right) \le b$ holds good for all x.
| Remark: In the meantime this approach has been also added by André Nicolas to his answer.
We will show that the sum $3\cos x+5\cos \left( x-\frac{\pi }{6}\right) $
can be written as $C\sin (x+\phi )$. From the difference formula for $\cos
\left( x-\frac{\pi }{6}\right) $ and using the values $\cos \frac{\pi }{6}=
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How do you divide a polynomial by a binomial of the form $ax^2+b$, where $a$ and $b$ are greater than one? I came across a question that asked me to divide $-2x^3+4x^2-3x+5$ by $4x^2+5$. Can anyone help me?
| Nobody seems to have posted the synthetic division route; I'll include it here for completeness.
We first monicize the divisor (i.e., set things up such that the divisor has leading coefficient $1$); thus, consider
$$\frac{-\frac12x^3+x^2-\frac34x+\frac54}{x^2+\frac54}$$
and set up the array
$$\begin{array}{r|cc|cc}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Are quartic minimal polynomials over $\mathbb{Q}$ always reducible over $\mathbb{F}_p$? This situation arose while studying biquadratic extensions.
Let $\mathbb{Q}(\alpha)$ is some biquadratic extension, with $m(x)$ the minimal polynomial of $\alpha$. Suppose that $m(x)\in\mathbb{Z}[x]$. Basic field theory and simple d... | An elementary argument, without any explicit reference to Galois theory:
Suppose the biquadratic extension is $K = \mathbb{Q}(\sqrt{B}, \sqrt{C})$, where $B$ and $C$ are integers (and none of $B$, $C$, $BC$ a rational square). Then any element $\alpha$ of $K$ has the form
$\alpha = a + b\sqrt{B} + c\sqrt{C} + d \sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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Simple probability problems This may be very simple, but I need to know correct answers.
Problem 1.
There are $6$ balls in the box - $2$ black, and $4$ white. We take $3$ balls from the box. What is the probability that we took exactly 1 black ball.
Problem 2.
What is the probability to score $7$ points when you throw ... | The critical distinction to make in problem 1 is whether you're sampling with or without replacement. This is, when you pick a ball out of the jar, do you note its color and put in back in, or keep it it. If it's the former case, you're sampling with replacement. In that case, the number of black balls in a sample o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/89297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Conditions for intersection of parabolas? What are the conditions for the existence of real solutions for the following equations:
$$\begin{align}
x^2&=a\cdot y+b\\
y^2&=c\cdot x+d\end{align}$$
where $a,b,c,d $ are real numbers.
These represent two parabolas; how might we find out the conditions for the existence of ... | Since these are parabolas, $a$ and $c$ must be nonzero. Let $x = a^{2/3} c^{1/3} X$ and $y = a^{1/3} c^{2/3} Y$. Under this scaling, the equations become
$X^2 = Y + B$ and $Y^2 = X + D$ where $B = b a^{-4/3} c^{-2/3}$ and
$D = d a^{-2/3} c^{-4/3}$. Now substituting $Y = B - X^2$ into $Y^2 = X + D$
we get the fourth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find $ n\geq1 $ such that 7 divides $n^n-3$ Find $ n\geq1 $ such that 7 divides $n^n-3$.
Here is what I found:
$ n\equiv 0 \mod7, n^n\equiv 0 \mod7,n^n-3\equiv -3 \mod7$ no solution.
$ n\equiv 1 \mod7, n^n\equiv 1 \mod7,n^n-3\equiv -2 \mod7 $ no solution.
$ n\equiv 2 \mod7, n^n\equiv 2^n \mod7, n^n-3\equiv 2^n-3 \mod7$... | $n^n - 3$ is divisible by $7$ if and only if $n \equiv 5\mod 42$ or $n \equiv 31 \mod 42$.
If $n \equiv 5 \mod 42$, then $n \equiv 5 \mod 7$. Thus $n^n \equiv 5^n \equiv 5^{n-5} \cdot 5^5 \equiv 5^5 \equiv 3 \mod 7$, because $n-5$ is divisible by $6$.
If $n \equiv 31 \mod 42$, then $n \equiv 31 \equiv 3 \mod 7$. As bef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Two players alternately flip a coin; what is the probability of winning by getting a head?
Two players, $A$ and $B$, alternately and independently flip a coin and the first player to get a head wins. Assume player $A$ flips first. If the coin is fair, what is the probability that $A$ wins?
So $A$ only flips on odd t... | The probability that the first head occurs on toss $n$ is $2^{-n}$, so the probability that the first head happens on an odd $n$ is
$$
\sum_{k=0}^{\infty}2^{-(2k+1)}=\frac{1}{2}\frac{1}{1-1/4}=2/3
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/94331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 3
} |
Solving a $1^\infty$ indeterminate form. I'm preparing for my calculus exam and I can't solve this limit:
$$\lim_{x\rightarrow\infty}\left(\frac{1+\tan(1/x)}{1-\tan(1/x)}\right)^x$$
The limit tends to $1^\infty$, which is indeterminate.
I've tried several things and I couldn't solve it.
Any idea? Thanks in advance.
| Asymptotics ...
$$\begin{align}
\operatorname{tan} \biggl(\frac{1}{x}\biggr) &= \frac{1}{x} + \frac{1}{3 x^{3}} + \frac{2}{15 x^{5}} + O \Bigl(x^{(-6)}\Bigr)\\
1 + \operatorname{tan} \biggl(\frac{1}{x}\biggr) &= 1 + \frac{1}{x} + \frac{1}{3 x^{3}} + \frac{2}{15 x^{5}} + O \Bigl(x^{(-6)}\Bigr)\\
1 - \operatorname{tan} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/98737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
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Which is the easiest way to evaluate $\int \limits_{0}^{\pi/2} (\sqrt{\tan x} +\sqrt{\cot x})$?
Which is the easiest way to evaluate $\int \limits_{0}^{\pi/2} (\sqrt{\tan x} +\sqrt{\cot x})$?
I have reduced this problem to $$ 2\int_0^{\pi/2} \sqrt{\tan x} \ dx$$
but now, evaluating this integral is giving me some p... | $${\int_0^{\frac{\pi}{2}} \sqrt{\tan x}dx + \sqrt{\cot x}dx}$$
$$={\int_0^{\frac{\pi}{2}}\frac{\sin x + \cos x}{\sqrt{\sin x \cos x}}dx = \int_0^{\frac{\pi}{2}} \frac{\sin x + \cos x}{\frac{\sqrt{2\sin{x}\cos{x}}}{\sqrt{2}}}dx = \sqrt{2}\int_0^{\frac{\pi}{2}} \frac{ \sin{x} + \cos{x}}{\sqrt{1 - (1 - 2 \sin{x} \cos{x})}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 8,
"answer_id": 2
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Partial derivative of an implicit function I am trying to solve this question (sorry if the translation is a bit vague):
$Z(x,y)$ is an implicit function of $x$ and $y$ given in the form of
$$x^3z^2+\frac{2}{9}y^2\sin(z) = xyz$$
in the neighborhood of $x=2, y=\pi, z=\frac{\pi}{6}$.
Find $Z_x$ and $Z_y$ at $(x,y) = (2... | Note: This answer was posted before the correction. However, the same idea still works.
I assume the given equation is
$$x^3z^2 + \frac{2}{9}y^2\sin z = \frac{\pi^2}{3}.$$
Let's now regard $z = Z(x,y)$ as a function of $x$ and $y$ and implicitly (partially) differentiate with respect to $x$:
$$\frac{\partial}{\partial... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Basic Epsilon-N proof clarification It's been a couple years since I've done analysis, so I was hoping someone could point out any possible flaws I have in the following proof.
Prove $\lim_{n\to\infty}\frac{2^n}{n!} = 0$.
For $n > 2$, we have: $\lim_{n\to\infty}\frac{2^n}{n!} = \lim_{n\to\infty}\frac{2^2}{2!}\times\fra... | A couple of things: first of all, writing $$\lim_{n \to \infty}\frac{2}{n} = 0 \ \ \Rightarrow \ \lim_{n\to\infty}\frac{2^2}{2!}\times\frac{2}{3}\times\frac{2}{4}\times\frac{2}{5}\times...\times\frac{2}{n} = 0$$ isn't completely rigorous, unless you observe that all the factors before $\frac{2}{n}$ are $< 1$ (except f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/103626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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find all positive integers satisfying $2x^2 - y^{14} = 1$ The following problem was posted to usenet forum de.rec.denksport two weeks ago and no progress was made.
Find all positive integers $x$,$y$ satisfying the equation
$$2x^2 - y^{14} = 1$$
$(1,1)$ is a solution and i suspect it is the only one.
I tried some thing... | The equation $v^2=y^{12}-y^{10}+y^8-y^6+y^4-y^2+1$ has only trivial solutions $y=0,1$.
Define $f(x)=1+x+x^2+x^3+x^4+x^5+x^6$ so that the right hand side is $f(-y^2)$. For $x>5$ we have $$(16x^3+8x^2+6x+5)^2<256 f(x)< (16x^3+8x^2+6x+6)^2,$$ while for $x<-4$ we get $$(16x^3+8x^2+6x+5)^2<256 f(x)< (16x^3+8x^2+6x+4)^2.$$
C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Evaluate fraction of sum So i have to evaluate this sum:
$\displaystyle \frac{1-2^{-2}+4^{-2}-5^{-2}+7^{-2}-8^{-2}+10^{-2}-11^{-2}+\cdots}{1+2^{-2}-4^{-2}-5^{-2}+7^{-2}+8^{-2}-10^{-2}-11^{-2}+\cdots}$
it has the form :
$\displaystyle \frac{\sum^{\infty}_0 [(3n+1)^{-2}-(3n+2)^{-2}]}{\sum^{\infty}_0 (-1)^n[(3n+1)^{-2}+(3... | First, let $A = 1-\frac{1}{2^2} + \frac{1}{4^2}-\frac{1}{5^2}+... $ and $B = 1 +\frac{1}{2^2}-\frac{1}{4^2}-\frac{1}{5^2}+... $
Notice that the both A and B converges absolutely (by comparison test). Thus, limit law applies: $B-A=2\times\frac{1}{2^2}-2\times\frac{1}{4^2}-2\times\frac{1}{8^2}-2\times\frac{1}{10^2}+... ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Compute $\int_{0}^{\frac{\pi}{2}}\frac{\sin^2x\cos x}{\sin x+\cos x}dx$ I'd like your help with computing the following integral:$$ \int_{0}^{\frac{\pi}{2}}\frac{\sin^2x\cos x}{\sin x+\cos x}dx.$$ I used $t=\sin x$ and got that $$\int_{0}^{\frac{\pi}{2}}\frac{\sin^2x\cos x}{\sin x+\cos x}dx=\int_{0}^{1}\frac{t^2}{t+\sq... | The light: Denote $\displaystyle I = \int^{\pi/2}_0 \frac{ \sin^2 x \cos x}{\sin x + \cos x} dx .$ Let $ u= \pi/2 -x $ to find $\displaystyle I = \int^{\pi/2}_0 \frac{\cos^2 x \sin x}{\sin x+\cos x} dx .$ Adding these two gives $\displaystyle 2I = \int^{\pi/2}_0 \frac{ \sin x \cos x (\sin x+ \cos x) }{\sin x + \cos x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Use mathematical induction to prove that for all integers $n \ge 2$, $2^{3n}-1$ is not prime I had a homework due yesterday with this problem.
The TA did the problem last week in discussion but I didn't understand it.
She pulled out a $7k$ almost immediately, and I have no idea from where.
It was like, it wasn't prime ... | \begin{align*}
2^{3n}−1 & =(2^3)^n−1\\
& =8^n−1^n\\
& =(8−1)(8^{n−1} \cdot 1^0+8^{n−2} \cdot 1^1+⋯+8^2 \cdot 1^{n−3}+8^1 \cdot 1^{n−2}+8^0 \cdot 1^{n−1})\\
& =7\sum(8^{n−1}+8^{n−2}+⋯+8^2+8+1)\\
& =7\sum_{k=0}^{n−1} 8^k
\end{align*}
\begin{align*}
2^{3n}−1 & =(2^3)^n−1\\
& = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 6
} |
Solving the exponential equation: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ I have this exponential equation that I don't know how to solve:
$3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ with $x \in \mathbb{R}$
I tried to factor out a term, but it does not help. Also, I noticed that:
$2 \cdot 9^... | $$
\begin{eqnarray}
0 &=& 3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1}
\\&=& 12 \cdot (2^x)^2 - 35 \cdot (2^x3^x) + 18 \cdot (3^x)^2
\\&=& 12 t^2 - 35 st + 18 s^2
\qquad\text{for}\qquad s=3^x,~t=2^x
\\&=& (3t - 2s)(4t - 9s)
\\
\implies&&
s=\frac32t \quad\text{or}\quad
s=\frac49t
\\&& s=\left(\frac23... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
find shortest distance from origin to parabolic form suppose that,we have given following parabolic arc
$\sqrt{x}+\sqrt{y}=\sqrt{a}$
we are trying to find shortest distance from origin to this line, i think that if we rewrite it as $\sqrt{y}=\sqrt{a}-\sqrt{x}$, then
$y=a-2\sqrt{ax}-x$ or if we rearrange terms... | One asks that the vector $(x,y)$ is orthogonal to the tangent at $(x,y)$ of the curve. Thus, assume that $(x,y)$ and $(x+h,y+k)$ are on the curve and that $h,k\to0$. Then
$$
\sqrt{x}+\sqrt{y}=\sqrt{a}=\sqrt{x+h}+\sqrt{y+k}=\sqrt{x}\sqrt{1+h/x}+\sqrt{y}\sqrt{1+k/y}
$$
hence
$$
\sqrt{x+h}+\sqrt{y+k}=\sqrt{x}+h/(2\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Finding domain of $\sqrt{ \frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} }$ How can I find the domain of:
$$\sqrt{ \frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} }$$
I think the hard part will be to find:
$$\frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} \ge 0$$
So far I have: not sure how to preceed:
$... | The first thing you should note is that the expression
$$\Phi=\frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} $$
is undefined at any zero of the denominator.
So, the points $x=\pm \sqrt 2$, $\pm 2$, and $\pm\sqrt 6$ are
not in the domain of $\sqrt\Phi$.
Now, to find the domain of $\sqrt\Phi$, we need to find wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Finding the outward flux through a sphere Problem:
Find the flux of of the field $F$ across the portion of the sphere $x^2 + y^2 + z^2 = a^2$ in the first octant in the direction away from the origin, when $F = zx\hat{i} + zy\hat{j} + z^2\hat{k}$.
| The way you calculate the flux of $F$ across the surface $S$ is by using a parametrization $r(s,t)$ of $S$ and then
$$
\int\!\!\!\!\int_S F\cdot n\, dS =
\int\!\!\!\!\int_D F(r(s,t))\cdot (r_s\times r_t)\, dsdt,
$$
where the double integral on the right is calculated on the domain $D$ of the parametrization $r$.
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show $4 \cos^2{\frac{\pi}{5}} - 2 \cos{\frac{\pi}{5}} -1 = 0$ Show
$$4 \cos^2{\frac{\pi}{5}} - 2 \cos{\frac{\pi}{5}} -1 = 0$$
The hint says "note $\sin{\frac{3\pi}{5}} = \sin{\frac{2\pi}{5}}$" and "use double/triple angle or otherwise"
So I have
$$4 \cos^2{\frac{\pi}{5}} - 2 (2 \cos^2{\frac{\pi}{10}} - 1) - 1$$
$$4 \... | The hint says to use the following:
$$\sin 3x = 3 \sin x - 4 \sin^3x$$
and
$$ \sin 2x = 2 \sin x \cos x$$
So if $x = \frac{\pi}{5}$, then we have, using $\sin 3x = \sin 2x$ that
$$3 \sin x - 4 \sin^3 x = 2 \sin x \cos x $$
Since $\sin x \neq 0$, cancel it, and use $\sin^2 x = 1 - \cos ^2x$.
For a geometric way to find ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
limit law and product of matrices Are there two $ n\times n $ matrices $A$ and $B$ such that $ \lim_{m\to\infty} A^m$ and $ \lim_{m\to\infty} B^m $ both exists but $ \lim_{m\to\infty} (A \cdot B)^m $ doesn't ?
| Consider
$$
A = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0\end{pmatrix} \qquad B = A^T
$$
Then $\forall m \geqslant 3$, $A^m = B^m = 0$, but since
$$
A \cdot B = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 2\end{pmatrix}
$$
$$
\lim_{m \to \infty} (A \cdot B)^m = \lim_{m \to \infty} \beg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
A problem about parametric integral How to solve the following integral.
$I(\theta) = \int_0^{\pi}\ln(1+\theta \cos x)dx$ where $|\theta|<1$
| Differentiating under the integral sign yields: $$ I'(\theta) = \int^{\pi}_0 \frac{\cos x}{1+ \theta \cos x} dx .$$
So $$\theta I'(\theta) = \int^{\pi}_0 1 - \frac{1}{1+\theta \cos x} dx= \pi - \int^{\pi}_0 \frac{1}{1+\theta \cos x} dx$$
To deal with the last integral, consider $$
\begin{aligned} I & = \int_{0}^{\pi} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.