qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
4,040,301 | <p>If <span class="math-container">$\lim_{|x| \to \infty} g(x)/x = \infty$</span>, Prove that <span class="math-container">$\{g(x)\mid x \in \mathbb{R}\} = \mathbb{R}.$</span></p>
| Reveillark | 122,262 | <p>Let <span class="math-container">$M>0$</span>.</p>
<p>By assumption, there is some <span class="math-container">$N>1$</span> such that <span class="math-container">$x\ge N$</span> implies <span class="math-container">$\frac{g(x)}{x}>M$</span>. For such <span class="math-container">$x$</span>, <span class="m... |
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | E.O. | 18,873 | <p><a href="http://www.khanacademy.org/" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8mCsM.jpg" alt="Khan Academy - A free world-class education for anyone anywhere."></a></p>
|
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | J. M. ain't a mathematician | 498 | <p><a href="https://mathematica.stackexchange.com/"><img src="https://i.stack.imgur.com/hOf2P.png" alt="Help this community grow!"></a></p>
|
1,454,919 | <p>I am trying to understand derivative and I want to know intuitive and rigorous definitions for a curve and if derivative is lmited only to curves or not..</p>
| Yes | 155,328 | <p>In elementary calculus context, there is no need to rigorously define what it mean by "curve". Intuitively you can think of a curve as an arbitrary line that can be drawn in one stroke; the simplest curve is a straight line. </p>
<p>The concept of derivative originates from the problem of finding the slope of a cur... |
1,820,036 | <p>I'd be thankful if some could explain to me why the second equality is true...
I just can't figure it out. Maybe it's something really simple I am missing?</p>
<blockquote>
<p>$\displaystyle\lim_{\epsilon\to0}\frac{\det(Id+\epsilon H)-\det(Id)}{\epsilon}=\displaystyle\lim_{\epsilon\to0}\frac{1}{\epsilon}\left[\de... | Community | -1 | <p><strong>Hint</strong>:</p>
<p>$$\left|\begin{matrix}
1+\epsilon h_{11}&\epsilon h_{12}\\
\epsilon h_{21}&1+\epsilon h_{22}\\
\end{matrix}\right|=1+\epsilon h_{11}+\epsilon h_{22}+\epsilon^2\left(h_{11}h_{22}-h_{12}h_{21}\right).$$</p>
<p>Only the main diagonal generates terms in $\epsilon$. This generalize... |
2,221,807 | <p>I know that this question has been answered before, however I have not seen a response that satisfies me on whether my proof will work.</p>
<p><strong>Proof</strong></p>
<p>Suppose $A \cup B$ is a separation of $X$. Then WLOG $X-A=B$ and is finite, but this implies that $X-B$ is infinite thus $B$ is not a open set... | Henno Brandsma | 4,280 | <p>If $A \cup B$ is a disconnection of $X$, this means that $A$ and $B$ are non-empty, disjoint and both open and closed (if one is open, the other is automatically closed as its complement, so we assume both are open or both are closed, usually,and it follows that both sets are clopen).</p>
<p>But by definition the o... |
747,561 | <p>I'm having trouble figuring out the limits. What messes me up is that the limit approaches infinity. Usually it approaches a specific number. Is that a trick to solve problems like these? </p>
<p>So for example, use the root test to find convergence/divergence. (n!)^n/(n^n)^7. n=1 and it's to infinity </p>
| David | 119,775 | <p>${\root n\of{a_n}}=n!/n^7\to\infty$, so $\sum a_n$ diverges.</p>
|
2,042,257 | <p>I'm looking at the following differential equation:</p>
<p>$\frac{dx}{dt} = \frac{\sin^2 x - t^2}{t \cdot \sin(2x)}$</p>
<p>I rewrote it as</p>
<p>$(t \cdot \sin(2x))dx = (\sin^2x - t^2)dt$</p>
<p>$\Leftrightarrow (\underbrace{\sin^2 x - t^2}_{J(x,t)})dt + (\underbrace{-t \cdot \sin(2x)}_{I(x,t)}) dx = 0$</p>
<... | Community | -1 | <p>As $x^x$ is a continous function, $1^1=1$ and $2^2=4$, then there is an $x$ such that</p>
<p>$$x^x=2.$$</p>
<p>As the function is monotonic in this range, the solution is unique.</p>
<p>This number is irrational, otherwise let $x$ be the irreducible fraction $p/q$:</p>
<p>$$\left(\frac pq\right)^{p/q}=2$$ implie... |
1,428,377 | <p>So I was watching the show Numb3rs, and the math genius was teaching, and something he did just stumped me.</p>
<p>He was asking his class (more specifically a student) on which of the three cards is the car. The other two cards have an animal on them. Now, the student picked the middle card to begin with. So the c... | David | 59,737 | <p>The probability that you've picked the <em>wrong</em> card in the first place is 2/3. This happens to be the truth even when one of the cards is removed; it's just that 2/3 of the initial guesses are wrong.</p>
<p>So when you have the chance to change your mind, you do it, because in the "second round", 1/2 of the ... |
498,694 | <p>So, I'm learning limits right now in calculus class.</p>
<p>When $x$ approaches infinity, what does this expression approach?</p>
<p>$$\frac{(x^x)}{(x!)}$$</p>
<p>Why? Since, the bottom is $x!$, doesn't it mean that the bottom goes to zero faster, therefore the whole thing approaches 0?</p>
| Alex | 38,873 | <p>This ratio grows at the rate $e^x \sqrt{2 \pi x}(1+o(1))$ which tends to infininty as $x \to \infty$</p>
|
2,259,145 | <p>Let $f\colon (0,1]\to [-1,1]$ be a continuous function. Let us define a function $h$ by $h(x)=xf(x)$ for all $x$ belongs to $(0,1]$.
Prove that $h$ is uniformly continuous.</p>
<p>We know $f$ is uniformly continuous on $I$ if $f'(x)$ is bounded on $I$. Here $h'(x)= xf'(x) + f(x)$ and $f(x)$ is bounded here. How can... | Jack D'Aurizio | 44,121 | <p>I will try to give this question an actual meaning: at the moment, there are definition issues in $\frac{1}{x}+\frac{1}{x-1}+\ldots+1$ if $x\not\in\mathbb{N}$. We may start from the Weierstrass product for the <a href="https://en.wikipedia.org/wiki/Gamma_function" rel="nofollow noreferrer">$\Gamma$ function</a>:
$$ ... |
1,132,063 | <p>For $x=(x_j)_{j\in\mathbb N}\in \ell^1$ let</p>
<p>$$\|x\|=\sup_{n\in \mathbb N}\left \Vert \sum_{j=1}^{n}x_j\right\Vert$$</p>
<p>Show that $(\ell^1,\|\cdot\|)$ is a normed space, but it is not complete.</p>
<p>The first part was easy.</p>
<p>Now I try to find a sequence in $\ell^1$ such that it is a cauchy sequ... | Pedro | 23,350 | <p>A normed space is Banach if and only if whenever a series converges absolutely, it converges. Try to find a sequence $(x_n)$ with $$\sum \lVert x_n\rVert<\infty$$ but that cannot possibly converge. </p>
|
2,165,759 | <p>I am solving the following question</p>
<p>$$\int\frac{\sin x}{\sin^{3}x + \cos^{3}x}dx.$$</p>
<p>I have been able to reduce it to the following form by diving numerator and denominator by $\cos^{3}x$ and then substituting $\tan x$ for $t$ and am getting the following equation. Should Iis there any other way use p... | user326159 | 326,159 | <p>$\textbf{Hint.}$ Firstly, </p>
<p>$$f(t)=\frac{t}{t^3+1}=\frac{t}{(t+1)(t^2-t+1)}=\frac{t+1}{3(t^2-t+1)}-\frac{1}{3(t+1)}$$</p>
<p>The second integral is immediate (a logarithmic function). The first needs more work, but it can be reduced to the integral of a logaritmic plus an $\arctan$</p>
|
2,165,759 | <p>I am solving the following question</p>
<p>$$\int\frac{\sin x}{\sin^{3}x + \cos^{3}x}dx.$$</p>
<p>I have been able to reduce it to the following form by diving numerator and denominator by $\cos^{3}x$ and then substituting $\tan x$ for $t$ and am getting the following equation. Should Iis there any other way use p... | Jack D'Aurizio | 44,121 | <p>We have that $-1,\omega,\omega^{-1}$ are the roots of $t^3+1$. In particular, $\frac{t}{t^3+1}$ can be represented as
$$ \frac{t}{t^3+1} = \frac{A}{t+1}+\frac{B}{t-\omega}+\frac{C}{t-\omega^{-1}}$$
with $A+B+C=0$ and
$$ A = \lim_{t\to -1}\frac{t(t+1)}{t^3+1} = \lim_{t\to -1}\frac{t}{t^2-t+1} = -\frac{1}{3} $$
hence... |
221,712 | <p>I have two matrix <code>A</code> and <code>B</code> of equal dimensions see below. In <code>A</code> matrix I have the variables <code>a,b,c,d</code> which have direct correspondence with matrix <code>B</code> element by each row. In other words, for first row <code>{a, b, c, d}</code> we have <code>{2, 9, 6, 7}</co... | WReach | 142 | <p>Here is a way using <code>ReplaceAll</code> (<code>/.</code>):</p>
<pre><code>{a - d, b - a} /. MapThread[Rule, {A, B}, 2]
(* {{-5, 7}, {1, -7}, {8, -11}} *)
</code></pre>
|
2,222,215 | <p>Determine whether the difference of the following two series is convergent or not and Prove your answer$$
\sum_{n=1}^\infty \frac{1}{n} $$ and $$\sum_{n=1}^\infty \frac{1}{2n-1} $$</p>
<p>What i tried. I said that the difference of the two series is divergent. My proof is as follows. Find the difference of the tw... | John Bentin | 875 | <p>The question is not meaningful: The difference of the series is not defined since one (actually, each one) of them is not defined.</p>
<p>If, nevertheless, you insist on going ahead and "subtracting" them, then why not try it in the following artistic way? $$\sum_{n=1}^\infty \frac{1}{n}-\sum_{n=1}^\infty \frac{1}{... |
2,555,815 | <p><strong>Problem</strong></p>
<p>Let $a_{0}(n) = \frac{2n-1}{2n}$ and $a_{k+1}(n) = \frac{a_{k}(n)}{a_{k}(n+2^k)}$ for $k \geq 0.$</p>
<p>The first several terms in the series $a_k(1)$ for $k \geq 0$ are:</p>
<p>$$\frac{1}{2}, \, \frac{1/2}{3/4}, \, \frac{\frac{1}{2}/\frac{3}{4}}{\frac{5}{6}/\frac{7}{8}}, \, \frac... | Kelenner | 159,886 | <p>Only a remark, not a complete answer.</p>
<p>For $q\in \mathbb{N}$, put $s_2(q)=$ the sum of the digits of the base two expansion of $q$, ie $s_2(3)=2$, $s_2(4)=1$, etc. The following formula can be proven by induction :</p>
<p>$$a_m(n)=\prod_{0\leq q<2^m}\left(1-\frac{1}{2n+2q}\right)^{(-1)^{s_2(q)}}$$ </p>
<... |
796,199 | <p>As far as I know, Brent's method for root finding is said to have superlinear convergence, but I haven't been able to find any more concrete information.</p>
<p>Is its convergence rate known to be at least bounded between some known values?</p>
<p>What is a good bibliographic reference for that?</p>
<p>[EDIT]</p>... | Simply Beautiful Art | 272,831 | <h2>Corrected answer:</h2>
<p>After a lot of testing on my own time, I noticed a particular anomaly when running Brent's method to high precisions. <strong>Brent's method never attains an order of convergence of <span class="math-container">$\mu\approx1.839$</span></strong>. In fact it doesn't attain an order of conver... |
3,612,351 | <p>It is given that a function f(x) satisfy:
<span class="math-container">$$f(x)=3f(x+1)-3f(x+2)\quad \text{ and } \quad f(3)=3^{1000}$$</span> then find value of <span class="math-container">$f(2019)$</span>.</p>
<p>I further wanted to ask that is there some general method to solve such equation. The method that I kn... | Pekisch | 735,184 | <p>This is a difference equation. In the case of a linear, constant coefficient difference equation we make the following guess:
<span class="math-container">$$ f(x) = r^x $$</span>
Then
<span class="math-container">$$ f(x+1) r\times r^x$$</span>
and
<span class="math-container">$$ f(x+2) = r^2\times r^x $$</span>
Repl... |
1,023,193 | <p>Proving this formula
$$
\pi^{2}
=\sum_{n\ =\ 0}^{\infty}\left[\,{1 \over \left(\,2n + 1 + a/3\,\right)^{2}}
+{1 \over \left(\, 2n + 1 - a/3\,\right)^{2}}\,\right]
$$
if $a$ an even integer number so that
$$
a \geq 4\quad\mbox{and}\quad{\rm gcd}\left(\,a,3\,\right) = 1
$$</p>
| Venus | 146,687 | <p>Alternatively, let's consider
$$f(a)=\sum_{n=0}^{\infty }\left(\frac{3}{2n+1-\frac{a}{3}}-\frac{3}{2n+1+\frac{a}{3}}\right)=9\sum_{n=0}^{\infty }\left(\frac{1}{6n+3-a}-\frac{1}{6n+3+a}\right)$$
so that our original sum is $f'(a)$.
$$\begin{align}
f(a)&=9\sum_{n=0}^{\infty }\int_0^1 \left(x^{6n+2-a}-x^{6n+2+a}\ri... |
24,055 | <p>Running this code:</p>
<pre><code>Histogram[{RandomVariate[NormalDistribution[1/4,0.12],100],
RandomVariate[NormalDistribution[3/4, 0.12], 100]},
Automatic, "Probability", PlotRange -> {{0, 1}, {0, 1}},
Frame -> True, PlotRangeClipping -> True,
FrameLabel -> {Style["x axis", 15], Style["probability... | Hubble07 | 7,009 | <p>When i ran your code on my system it was fine.
Version No: 9.0.0.0
Platform: Linux x86(32-bit)
So maybe its a windows problem.
Try Exporting the image and see if that 'y' is still missing in the exported image.</p>
|
24,055 | <p>Running this code:</p>
<pre><code>Histogram[{RandomVariate[NormalDistribution[1/4,0.12],100],
RandomVariate[NormalDistribution[3/4, 0.12], 100]},
Automatic, "Probability", PlotRange -> {{0, 1}, {0, 1}},
Frame -> True, PlotRangeClipping -> True,
FrameLabel -> {Style["x axis", 15], Style["probability... | a06e | 534 | <p>I got the following reply from Technical Support @ Wolfram:</p>
<blockquote>
<p>Hello - </p>
<p>Thank you for your email.</p>
<p>Our developers have created a report on this issue and are
investigating the issue.</p>
<p>If you need a workaround for this issue, you have a number of
possibilities... |
92,670 | <p>We're learning about domains and setbuilder notation in school at the moment, and I want to make sure what I did was right.</p>
<p>My thought process:
\begin{align*}
-\frac12|4x - 8| - 1 &< -1 \\
-\frac12|4x - 8| &< 0 \\
|4x - 8| &> 0
\end{align*}
$x =$ all real numbers.</p>
<p>{real numbers} :</... | Community | -1 | <p>First let's consider how absolute-value is defined:</p>
<p>$$
|a| =
\begin{cases}
a, & \text{if } a \geq 0,
\\ -a, &\text{if } a \lt 0.
\end{cases}
$$</p>
<p>Therefore,</p>
<p>$$4x-8 > +0\phantom{.}$$</p>
<p>$$4x-8 < -0.$$</p>
<p>Now, solve for $x$ to get the answer:</p>
<p>$$x > 2\phantom{.}... |
2,236,008 | <p>Suppose $Z$ is a Gaussian distribution $N(0,\sigma^2)$. Is there a formula of upper bound for $P(Z\in [a,b])$, or do we know this probability is integral with respect to $\sigma\in \mathbf{R}$?</p>
| Community | -1 | <p>If $b>a$ then the upper bound of integration is $b$. As area under a curve can be measured by integral and we know that the probability is some area under a curve, therefore we do know that probability is integral with respect to $x$.</p>
|
1,556,805 | <p>I'm working in a problem that involves the equation
$$
w(z)=\sqrt{1-z^{2}} \,\, .
$$</p>
<p>I already know that there're two branch points in this equation, namely $\pm 1$, so there's a Riemann surface covering the domain of the function where the branch cut is from the $-1$ to $1$, as shown in the figure below.</... | Yiorgos S. Smyrlis | 57,021 | <p>Your idea works fine, since $w$ can be defined as a holomorphic function in the unit disk (for example).</p>
<p>Use that fact that
$$
\sqrt{1-x}
= \sum_{k=0}^{\infty} (-1)^k\binom{1/2}{k}x^k=1+\sum_{k=1}^{\infty}\frac{(-1)^k(1/2)(1/2-1)\cdots (1/2-k+1)x^k}{k!} \\ =1+\sum_{k=1}^{\infty}\frac{(-1)^k(-1)(-3)\cdots (-2... |
1,181,123 | <blockquote>
<ol>
<li>Find the smallest positive integer such that $80-n$ and $80+n$ are prime numbers. </li>
<li>Find the smallest positive prime number such that $2002-n$ and $2002+n$ are prime numbers.</li>
</ol>
</blockquote>
<p>I cannot think of any way other than trying the prime numbers one by one,
li... | Joffan | 206,402 | <p>For question 2, note that the question asks for a prime number n, unlike question 1. The same reasoning applies as in <a href="https://math.stackexchange.com/users/83272/fermat">Fermat</a>'s <a href="https://math.stackexchange.com/a/1181141/206402">answer</a> in terms of mod 3 analysis:</p>
<p>$2002 \equiv 1 \bmod ... |
723,570 | <p>In the proof of Theorem 6.11, $\varphi$ is uniformly continuous and hence for arbitrary $\epsilon > 0$ we can pick $\delta > 0$ s.t. $\left|s-t\right| \leq \delta$ implies $\left|\varphi\left(s\right)-\varphi\left(t\right)\right|<\epsilon$. However, I do not understand why he claims that $\delta < \epsil... | orion | 137,195 | <p>The question is basically just a sieve, similar to looking for primes.</p>
<p>The period that visits all combinations of said numbers is long ($2\times 3\times 5 \times 7=210$) so whatever you do, it won't be much quicker than the brute method.</p>
<p>For instance, starting with 2 and 5, you have the ones that are... |
3,306,747 | <p>Here is my attempt </p>
<p>h = 3k -7 ----(1)</p>
<p>(h-1)^2 + (k -1)^2 = 10/4</p>
<p>(h-1)^2 + (3h - 8)^2 = 10/4</p>
<p>This second one doesn't working.Is my approch wrong?</p>
<p>P.S: Sorry for the typo.Also I assumed the center is C(h,k)</p>
| Zacky | 515,527 | <p><span class="math-container">$$S=2\int_{0}^{1}\frac{x}{1-x^2}\left(\frac{\pi^2}{2}-2\arcsin^2(x)\right)dx\overset{IBP}=-4\int_0^1 \frac{\arcsin x\ln(1-x^2)}{\sqrt{1-x^2}}dx$$</span>
<a href="https://math.stackexchange.com/questions/292468/fourier-series-of-log-sine-"><span class="math-container">$$\overset{x=\sin t}... |
207,040 | <p>Is there some way I can solve the following equation with <span class="math-container">$d-by-d$</span> matrices in Mathematica in reasonable time?</p>
<p><span class="math-container">$$AX+X'B=C$$</span></p>
<p>My solution below calls linsolve on <span class="math-container">$d^2,d^2$</span> matrix, which is too ex... | xzczd | 1,871 | <p>First of all, <code>PDE == nv1 + nv2 + nv3 + nv4</code> is obviously wrong, because there already exists a <code>==</code> in your <code>PDE</code>. This is easy to fix of course.</p>
<p>What's confusing is the <code>Power::infy</code> warning. I'm not sure why this pops up, maybe <code>NDSolve</code> fails to noti... |
3,992,495 | <blockquote>
<p><span class="math-container">$\displaystyle b_1=\left\lbrace\frac{12}{9},\frac{12}{9},2\right\rbrace^T,b_2=\{-18,-18,21\}^T$</span> and <span class="math-container">$\displaystyle v_1=\{-1,-1,2\}^T,v_2=\{3,3,-3\}^T$</span>. <span class="math-container">$b_1 \in \operatorname{Span}\{v_1,v_2\} \text{ and ... | ironX | 534,898 | <p>If <span class="math-container">$b_1 \in \text{Span}(v_1, v_2) $</span> and <span class="math-container">$b_2 \in \text{Span}(v_1, v_2)$</span>, then <span class="math-container">$\text{Span}(b_1, b_2) \subset \text{Span}(v_1, v_2)$</span>.</p>
<p><em>Proof:</em></p>
<p><span class="math-container">$b_1 \in \text{Sp... |
3,992,495 | <blockquote>
<p><span class="math-container">$\displaystyle b_1=\left\lbrace\frac{12}{9},\frac{12}{9},2\right\rbrace^T,b_2=\{-18,-18,21\}^T$</span> and <span class="math-container">$\displaystyle v_1=\{-1,-1,2\}^T,v_2=\{3,3,-3\}^T$</span>. <span class="math-container">$b_1 \in \operatorname{Span}\{v_1,v_2\} \text{ and ... | Ali Ashja' | 437,913 | <p>As @ironX & @pietro said, and also by definition of vector spaces, they contain the <span class="math-container">$Span$</span> of any subset of themselves. But for inverse, also as they said truly, you can't conclude it. Of course, if you are curious to find such condition:</p>
<p><br>If you have <span class="ma... |
1,289,626 | <blockquote>
<p>find the Range of $f(x) = |x-6|+x^2-1$</p>
</blockquote>
<p>$$ f(x) = |x-6|+x^2-1 =\left\{
\begin{array}{c}
x^2+x-7,& x>0 .....(b) \\
5,& x=0 .....(a) \\
x^2-x+5,& x<0 ......(c)
\end{array}
\right.
$$</p>
<p>from eq (b) i got $$f(x)= \left(x+\frac12\right)^2-\frac{29}4 \ge-\fra... | wythagoras | 236,048 | <p>You should have $x-6<0$, $x-6=0$ and $x-6>0$ respectively. Always look to the entire expression within the absolute value. </p>
<p>Oh, and another thing: While finding such minimum, you need to check whether it is in the domain. For example, you have $x>0$ (should be $x>6$) for (b), but the minimum is g... |
3,371,888 | <p><span class="math-container">$$\left(\!\!{{a+b}\choose k}\!\!\right)= \sum_{j=0}^k \left(\!\!{a\choose j}\!\!\right) \cdot \left(\!\!{b\choose {k-j}}\!\!\right)$$</span></p>
<p>I am quite confused about the case of multichoose. I was able to prove this equation if only "n choose k" form was used as both sides would... | epi163sqrt | 132,007 | <p>This binomial identity is an instance of the <em><a href="https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Chu%E2%80%93Vandermonde_identity" rel="nofollow noreferrer">Chu-Vandermonde identity</a></em>.</p>
<blockquote>
<p>We start with the right-hand side. We obtain
<span class="math-container">\be... |
1,676,848 | <blockquote>
<p>Given the series </p>
<p>$$ \sum_{n=1}^{\infty} \frac{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)x^n}{n!} \quad \quad k \geq 1 $$
Find the interval of convergence.</p>
</blockquote>
<p>I started by applying the Ratio test</p>
<p>$$
\lim_{n\to \infty}\left|\frac{k(k+1)(k+2)\cdot \cdot \cdot (k + ... | robjohn | 13,854 | <p>Note that for $k\ge1$, we have
$$
\frac{k(k+1)\cdots(k+n-1)}{n!}=\frac k1\frac{k+1}2\cdots\frac{k+n-1}{n}\ge1
$$
Thus, for $|x|=1$, the terms do not go to $0$.</p>
|
2,316,448 | <p>I was working on the infinite sum
$$\sum_{x=1}^\infty \frac{1}{x(2x+1)}$$
and I used partial fractions to split up the fraction
$$\frac{1}{x(2x+1)}=\frac{1}{x}-\frac{2}{2x+1}$$
and then I wrote out the sum in expanded form:
$$1-\frac{2}{3}+\frac{1}{2}-\frac{2}{5}+\frac{1}{3}-\frac{2}{7}+...$$
and then rearranged it... | kvicente | 452,277 | <p>This is one of the most astonishing things in mathematics: <strong>every conditionally convergent series</strong> (meaning: that converges but is not absolutely convergent, such as the alternate harmonic series you mentioned) <strong>can be conveniently rearranged to converge to any arbitrary real number, or just di... |
216,171 | <p>Basically, I have a set of differential equations that I need to solve for exactly 100 different initial conditions (given as lists for each initial condition), and then plot each solution.</p>
<p>Here is some sample code where I have set vrad, vtan, and deltaR (arrays of initial conditions) to an array of length t... | Mark R | 65,931 | <p>I think this will do what you want:</p>
<pre><code>s = NDSolve[{r''[t] ==
r[t]*\[Phi]'[t]^2 - 1/r[t], \[Phi]'[t] == #[[4]]/r[t]^2, \[Phi][
0] == #[[1]], r[0] == #[[2]],
r'[0] == #[[3]]}, {r, \[Phi]}, {t, 0, 200}] & /@
Transpose[{vTangential/r0, r0, vRadial, L}]
</code></pre>
<p>Your curr... |
1,223,823 | <p>How can one simplify
$$\arctan\left(\frac{1}{\tan \alpha}\right)?$$
$0<α<\dfrac{\pi}{2}.$ Here is what I tried so far,
$$\arctan\left(\dfrac{1}{\tan \alpha}\right)=θ$$ for some θ.
$$\frac{1}{\tan \alpha}=\tan(θ)$$</p>
<p>I didn't know what to do next because there is no significant relationship between ${θ}$... | Narasimham | 95,860 | <p>Recognize the complementary angle relation:</p>
<p>$$\arctan\left(\frac{1}{\tanα}\right) =\arctan( \tan (\pi/2-\alpha) ) $$</p>
<p>$ \rightarrow (\pi/2 - \alpha), (3\pi/2 - \alpha), $ plus co-terminal angles.</p>
|
346,432 | <p>I will think of <span class="math-container">$ \mathbb{R}^{n+m}$</span> as <span class="math-container">$\mathbb{R}^n \times \mathbb{R}^m$</span>.</p>
<p>Let <span class="math-container">$ V \subset \mathbb{R}^{n+m}$</span> be open and <span class="math-container">$g:V \to U \subset \mathbb{R}^{n+m} $</span> be a... | Behnam Esmayli | 91,442 | <p>I have the answer here: <a href="https://mathoverflow.net/questions/350952/fubinis-theorem-on-arbitrary-foliations">Fubini's Theorem on Arbitrary Foliations</a></p>
<p><span class="math-container">$$\int_U f = \int_{U_{\eta_0}} \left(\int_{U_\xi} f(\xi,\eta) \frac{|\det DG_{U_\xi} (\xi,\eta)| \cdot |\det DG_{U_... |
2,616,663 | <p>For this proof, after I convert the definite integral into the riemann sum definition, is it just enough to say $\Delta(x)$ = $\frac{b-a}{n}$ and since b = a, the $\Delta(x)$ becomes 0, thus, making everything else equal to 0, since everything else is being multiplied by zero?</p>
| Renji Rodrigo | 522,531 | <p>We can use also the empty sum definition
$$\sum^0_{k=1} f(k)=0 $$ for every f.</p>
<p>One partition is in the form
$a=t_0<t_1< \ldots <t_n=b$
if $a=b$ then $n=0$.</p>
<p>Let it be the degenerate interval $[a,a]=\{a\}$, we have $t_{0}=a$ and $t_{n}=a$, we should have $t_{1}>t_{0}=a$ but we don't have... |
907,879 | <p>Calculate the limit $\lim\limits_{x\to\infty} (a^x+b^x-c^x)^{\frac{1}{x}}$ where $a>b>c>0$.</p>
<p>First,
$$\exp\left( \lim\limits_{x\to\infty} \frac{\ln(a^x+b^x-c^x)}{x} \right)$$</p>
<p>Next,
$$\lim\limits_{x\to\infty} a^x + b^x - c^x = \lim\limits_{x\to\infty} a^x \left[1 + (b/a)^x - (c/a)^x \right] = ... | André Nicolas | 6,312 | <p>The problem can be solved by Squeezing, using minimal algebraic manipulation. Note that for positive $x$ we have
$$a^x\lt a^x+b^x-c^x\lt 2a^x.$$
Now take the $x$-th roots. We get
$$a\lt (a^x+b^x-c^x)^{1/x}\lt 2^{1/x}a.$$
But $\lim_{x\to\infty} 2^{1/x}=1$, and it's over. </p>
|
702,804 | <p>I just need a sanity check, been thinking about this all morning.</p>
<p>If we use the Mean Value Theorem on a function over the infinite interval (suppose the function's domain is unbounded), i.e.</p>
<p>$$M=\lim\limits_{T \to \infty} \dfrac{1}{2T}\int_{-T}^{T} \text{dt} f(t)$$</p>
<p>There is no way that M can ... | ketan | 122,095 | <p>The solution 1 xpands the Function $y = \dfrac {1}{2-x}$ about $x=1$. whereas the second solution expands $y = \dfrac {1}{2-x}$ about $x=0$.</p>
|
25,137 | <p>I want to find an intuitive analogy to explain how binary addition (more precise: an adder circuit in a computer) works. The point here is to explain the abstract process of <em>adding</em> something by comparing it to something that isn't abstract itself.</p>
<p>In principle: An everyday object or an action that is... | Wyck | 13,481 | <p>I think if you've played Monopoly you understand that once you get 10 one-dollar bills, you'd rather trade them in for a ten-dollar-bill. And once you get 10 ten-dollar bills, you'd rather trade them in for a hundred-dollar-bill. You're trying to minimize the number of bills you have to manage. It's easier to kno... |
1,021,753 | <p>Any idea on how to compute the expected value of product of Ito's Integral with two different upper limit?</p>
<p>For example:
$$\mathbb{E}\left[\int_0^r f(t)\,dB(t) \int_0^s f(t)\,dB(t)\right]$$</p>
<p>I only know how to compute when the upper limit r and s are the same...but don't know how when r and s are diffe... | sds | 37,092 | <p>Your given rotation $R(l,\theta)$ around line $l$ by angle $\theta$ is a composition of two symmetries wrt planes $p_1$ and $p_2$ which intersect along $l=p_1\cap p_2$ at angle $\theta/2$:</p>
<p>$$ R(l,\theta) = S(p_1)\circ S(p_2) = S(p_1)\circ S(p)\circ S(p)\circ S(p_2) $$ </p>
<p>where $p$ is the plane you are... |
501,660 | <p>In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?</p>
<p>Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.</p>... | Muralidhar G | 244,309 | <p>Approximate the Taylor series.
In Taylor series we have to use the angle in radians and by converting it into degrees and by making some approximations we can get a simple formulas like
$\sin X = 0.017*X$ for $X<33$ degrees and
$\sin X = 0.016*X$ for $33 < X < 45$</p>
<p>$\cos X=1-0.000145 X^2$ for $X<... |
501,660 | <p>In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?</p>
<p>Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.</p>... | MCCCS | 357,924 | <p>Bhaskara's approximation (<a href="https://en.wikipedia.org/wiki/Bhaskara_I's_sine_approximation_formula" rel="noreferrer">Wikipedia</a>) gives an approximation for <span class="math-container">$\sin x^\circ$</span> with less than <span class="math-container">$0.0016$</span> error for <span class="math-container... |
501,660 | <p>In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?</p>
<p>Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.</p>... | richard1941 | 133,895 | <p>I like continued fractions. For example, if x is in (-pi/2, pi/2), as mentioned above for power series, </p>
<p>sin(x) = x/1+ x^2/6+ x^2, which matches the first 8 terms of the Taylor series given above.</p>
<p>tan(x) = x/1- x^2/3- x^2/5- x^2/7. This also matches the first 8 terms of the Taylor series for t... |
351,642 | <p>So I'm proving that a group $G$ with order $112=2^4 \cdot 7$ is not simple. And I'm trying to do this in extreme detail :) </p>
<p>So, assume simple and reach contradiction. I've reached the point where I can conclude that $n_7=8$ and $n_2=7$. </p>
<p>I let $P, Q\in \mathrm{Syl}_2(G)$ and now dealing with cases th... | Mikko Korhonen | 17,384 | <p>If $G$ is a simple group, it must have exactly $7$ Sylow $2$-subgroups. Thus $G$ embeds into $S_7$, and in particular into $A_7$ since $G$ does not have a subgroup of index $2$. But the order $A_7$ is not divisible by $112$.</p>
<p>If you want to go along the lines of your original idea, you can rule out the case $... |
4,579,084 | <p>It was a new contributor's question. I answered, got my -1 again and then deleted. Then I asked myself. Then gave it up again. Actually I was gonna ask a different question NOW. When I pressed ask a question, to my surprise, the question I intended to ask yesterday was in the memory!</p>
<p>I wanted to evaluate the ... | Peter Leopold | 517,642 | <p>You illustrate one case where the permutation is not <span class="math-container">$i$</span>-orderly, but there is another where it is, if you take <span class="math-container">$m=1$</span> and <span class="math-container">$A_1=\{1,2\}$</span> and <span class="math-container">$B_1=\{3\}$</span>. What about <span cla... |
153,217 | <blockquote>
<p>Let $$f(x)=\frac{2x+1}{\sin(x)}$$ Find $f'(x).$ </p>
</blockquote>
<p>I used Quotient Rule <br>
$$\begin {align*}\frac{\sin(x)2-(2x+1)\cos(x)}{\sin^2(x)}\\
=\frac{3-2x\cos(x)}{\sin(x)} \end {align*}$$</p>
<p>Is that right? I don't know how to get the answer.
Please help me out, thanks.</p>
| Brian M. Scott | 12,042 | <p>You seem to have some serious problems with the algebra involved. Part of the problem is failure to use necessary parentheses: the result of applying the quotient rule is</p>
<p>$$\frac{2\sin x-(2x+1)\cos x}{\sin^2x}\;,$$</p>
<p>where the parentheses around $2x+1$ are absolutely necessary. If you choose to multipl... |
153,217 | <blockquote>
<p>Let $$f(x)=\frac{2x+1}{\sin(x)}$$ Find $f'(x).$ </p>
</blockquote>
<p>I used Quotient Rule <br>
$$\begin {align*}\frac{\sin(x)2-(2x+1)\cos(x)}{\sin^2(x)}\\
=\frac{3-2x\cos(x)}{\sin(x)} \end {align*}$$</p>
<p>Is that right? I don't know how to get the answer.
Please help me out, thanks.</p>
| Gigili | 181,853 | <p>$$f(x)=\frac{2x+1}{\sin(x)}$$</p>
<p>As for the derivative of such a fractional function:</p>
<p>$$f'(x)=\frac{(2x+1)'(\sin x)-(\sin x)'(2x+1)}{\sin^2 x}$$</p>
<p>Simplifying:</p>
<p>$$f'(x)=\frac{2(\sin x)-\cos x(2x+1)}{\sin^2 x}=\frac{2\sin x-2x\cos x-\cos x}{\sin^2 x}=2\csc x-(2x+1) \cot x \csc x$$</p>
|
625,821 | <p>$$\int^\infty_0\frac{1}{x^3+1}\,dx$$</p>
<p>The answer is $\frac{2\pi}{3\sqrt{3}}$.</p>
<p>How can I evaluate this integral?</p>
| GPerez | 118,574 | <p>In general, when you have $$\frac{Bx+C}{ax^2+bx+c} = \frac{Bx}{ax^2+bx+c} + \frac{C}{ax^2+bx+c}$$</p>
<p>Then the left addend can be easily integrated by multiplying by $2a/B$ (and dividing outside the integral sign). For the left addend, assuming we can't factorize it in $\mathbb R$, the procedure I'd use is to wr... |
1,869,564 | <p>i tried to derive logistic population model, and need to integrate this
$\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t$. here is my solution</p>
<p>$\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t=\int \frac{1}{k-N_t} dN_t=-\int \frac{1}{k-N_t}d{(k-N_t)}=-\ln\mid k-N_t\mid+C_1$. i think i have done something wrong he... | Piquito | 219,998 | <p>QUESTION.- Do you want your curves necessarily be all concave? If not, you have a nice example to add to the concave ones with the Witch of Agnesi, whose equation is $y=\frac{8a^3}{x^2+4a^2}$ where $a$ is the radius of the circle that generates the curve (so you have infinitely many examples).</p>
<p>In... |
1,079,356 | <p>My question can be summarized as:</p>
<blockquote>
<p>I want to prove that closed immersions are stable under base change.</p>
</blockquote>
<p>This is exercise II.3.11.a in Hartshorne's Algebraic Geometry. I researched this for about half a day. I consulted a number of books and online notes, but I found the pr... | Exodd | 161,426 | <p>Consider a cartesian diagram
$$\require{AMScd}
\begin{CD}
X^\prime @>>> X \\
@VVV @VVV \\
Y^\prime @>>> Y
\end{CD}$$
where $X\to Y$ is a closed immersion. Let's call $g$ the function $X'\to Y'$. </p>
<p>We know that, taking $Y_i$ an open affine covering of $Y$, and $X_i$, $Y_i'$ open affine cove... |
4,528,838 | <p>Find the general solution of the equation <span class="math-container">$$x^{(5)} + 2x^{(4)} + 2x^{(3)} + 4x'' + x' + 2x = 100e^{-2t}.$$</span></p>
<p>I don't understand how solve such tasks. I know that I should solve <span class="math-container">$x^{(5)} + 2x^{(4)} + 2x^{(3)} + 4x'' + x' + 2x =0$</span> and then us... | user577215664 | 475,762 | <p>Hint:
<span class="math-container">$$x^{(5)} + 2x^{(4)} + 2x^{(3)} + 4x'' + x' + 2x = 100e^{-2t}.$$</span>
It's easier to solve this:
<span class="math-container">$$y'''' + 2y'' + y= 100e^{-2t}$$</span>
Where <span class="math-container">$y=x' + 2x $</span>.
<span class="math-container">$$y'''' + 2y'' + y= 0$$</span... |
2,293,147 | <p>I was trying to solve this ODE $\frac{dy}{dx} = c_{1} + c_{2}y + \frac{c_{3}}{y} , y(0) = c , c >0$.</p>
<p>where $c_{1},c_{2},c_{3}$ are three real numbers say $c_{1} < 0,c_{2},c_{3} > 0$.</p>
<p>I thought of using separation of variables giving me $x = \int(\frac{y}{c_{1}y+c_{2}y^2+c_{3}})dy + c$.</p>
... | BAYMAX | 270,320 | <p>Yes I agree what Yves say that I must get some type $\log$ and derivative of $\arctan$ function , but I am curious that when I consider the above integral which i want to find $\int(\frac{y}{c_{1}y+c_{2}y^2+c_{3}})dy $ then MATLAB returns </p>
<p>where for simplification I have taken $c_{1} = a ,c_{2} = b , c_{3} =... |
3,909,972 | <p>I used Photomath and Microsoft Math to compute an equation, but they gave me two different results (-411 and -411/38) Why did that happen and which is the correct answer?</p>
<p><a href="https://i.stack.imgur.com/egt5A.jpg" rel="nofollow noreferrer">https://i.stack.imgur.com/egt5A.jpg</a>
<a href="https://i.stack.im... | Community | -1 | <p>Both calculators gave you the correct answer: <em>false</em>.</p>
|
396,440 | <p>Suppose we have the function $$f(x) = \frac{x}{p} + \frac{b}{q} - x^{\frac{1}{p}}b^{\frac{1}{q}}$$ where $x,b \geq 0 \land p,q > 1 \land \frac{1}{p}+\frac{1}{q} = 1$</p>
<p>I am trying to show that $b$ is the absolute minimum of $f$. </p>
<p>I proceeded as follows:</p>
<p>$$\frac{df(x)}{dx} = \frac{1}{p} - \fr... | Inceptio | 63,477 | <p>Let $\dfrac{1}{p}=a$ and $\dfrac{1}{q}=k$</p>
<p>$ax+bk-x^a \cdot b^k=f(x) $</p>
<p>$\dfrac{x + \dots x_{ath}+ b+ \dots b_{kth}}{a+k} \ge (x^ab^k)^{1/(a+k)}$ (By <a href="http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means" rel="nofollow">AM-GM inequality)</a></p>
<p>$x+ \dots x (a$ times)... |
223,582 | <p>Maps $g$ maps $\left\{1,2,3,4,5\right\}$ onto $\left\{11,12,13,14\right\}$ and $g(1)\neq g(2)$. How many g are there.</p>
<p><strong>My answer</strong>:
I transformed the question to a easy-understand way and find out the solution.
Consider there are five children and four seats. Two of them are willing sitting to... | Jack D'Aurizio | 44,121 | <p>Intersect the circle having $AC$ as a diameter with the initial circle: you will find the two points $D,D'$ such that $CD$ and $CD'$ are tangent to the initial circle. This comes from the fact that the circle is the locus of points that "see" any diameter under an angle equal to $\frac{\pi}{2}$.</p>
|
1,363,144 | <p>Given a cubic polynomial $f(x) = ax^{3} + bx^{2} + cx +d$ with arbitrary real coefficients and $a\neq 0$. Is there an easy test to determine when all the real roots of $f$ are negative?</p>
<p>The Routh-Hurwitz Criterion gives a condition for roots lying in the open left half-plane for an arbitrary polynomial with ... | P Vanchinathan | 28,915 | <p>If you are interested in only the roots you can normalize and take $a=1$. Then a necessary condition is that $b,c,d>0$. As it has 3 negative roots its two turning points should be negative too. That is $f'(x)= 3x^2-2bx+c$ should have real negative roots, which can be easily translated to a condition on the discri... |
4,045,755 | <blockquote>
<p>If <span class="math-container">$p$</span> is a prime then all the non trivial subgroups of <span class="math-container">$G$</span> with <span class="math-container">$\lvert G\rvert=p^2$</span> are cyclic.</p>
</blockquote>
<p>I tried looking online where does this result come from, but could not find a... | GreginGre | 447,764 | <p>You do not need the classification of groups of order <span class="math-container">$p^2$</span>. Just use Lagrange theorem to conclude that a nontrivial subgroup (that is, different of the trivial one and <span class="math-container">$G$</span>) has order <span class="math-container">$p$</span>.</p>
<p>Now, groups o... |
33,622 | <p>I am looking for differentiable functions $f$ from the unit interval to itself that satisfy the following equation $\forall\:p \in \left( 0,1 \right)$:</p>
<p>$$1-p-f(f(p))-f(p)f'(f(p))=0$$</p>
<p>Is there a way to use <em>Mathematica</em> to solve such equations?<br>
<code>DSolve</code> is of course unable to han... | jlperla | 9,151 | <p>To answer the question of: does mathematica have facilities for this sort of thing. I think there may be 2 parts:
1) Does mathematica have facilities for general function equations (forgetting even the differential)? I think the answer is no in general. There are certain (recurrence) equations it can solve with <... |
1,289,868 | <p>EDIT (<em>now asking how to write $F$ as distributions, instead of writing the integral in terms of distributions</em>): </p>
<p>Let $F$ be the distribution defined by its action on a test function $\phi$ as </p>
<p>\begin{equation*}
F(\phi)=\int_{\pi}^{2\pi}x\phi(x)dx.
\end{equation*}</p>
<p>How would you write... | Nikita Evseev | 23,566 | <p>Suspect one could not find continuous $g(x)$.
I state that $g(x) = x\cdot H(x-\pi)\cdot H(2\pi-x)$. Namely $g(x) = x$ if $x\in[\pi, 2\pi]$ and $g(x)=0$ for $x\in R\setminus [\pi, 2\pi]$. So
$$
\int_{\pi}^{2\pi}x\phi(x)dx = \int_{-\infty}^{+\infty}g(x)\phi(x)dx.
$$</p>
|
481,421 | <p>Find the limit of:
$$\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$$</p>
| Boris Novikov | 62,565 | <p>Since $\cos t \sim 1-t^2/2$ from Taylor series, then
$$
\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}= \lim_{x\to\infty}{\frac{\frac{1}{2x^2}}{\frac{4}{2x^2}}}=1/4
$$</p>
|
481,421 | <p>Find the limit of:
$$\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$$</p>
| obataku | 54,050 | <p>Rewriting $\cos(2/x)=2\cos^2(1/x)-1$ we have:$$\begin{align*}\lim_{x\to\infty}\frac{\cos(1/x)-1}{\cos(2/x)-1}&=\lim_{x\to\infty}\frac{\cos(1/x)-1}{2\cos^2(1/x)-2}\\&=\frac12\lim_{x\to\infty}\frac{\cos(1/x)-1}{\cos(1/x)^2-1}\\&=\frac12\lim_{x\to\infty}\frac{\cos(1/x)-1}{(\cos(1/x)+1)(\cos(1/x)-1)}\\&=... |
481,421 | <p>Find the limit of:
$$\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$$</p>
| DonAntonio | 31,254 | <p>What about a little l'Hospital?</p>
<p>$$\lim_{x\to\infty}\frac{\cos\frac1x-1}{\cos\frac2x-1}\stackrel{\text{l'H}}=\lim_{x\to\infty}\frac{\frac1{x^2}\sin\frac1x}{\frac2{x^2}\sin\frac2x}\stackrel{\text{l'H}}=\frac12\lim_{x\to\infty}\frac{-\frac1{x^2}\cos\frac1x}{-\frac2{x^2}\cos\frac2x}=\frac12\cdot\frac12\cdot\frac... |
481,421 | <p>Find the limit of:
$$\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$$</p>
| mnsh | 58,529 | <p>$$\lim_{x \to \infty} \frac{\cos \frac1x - 1}{\cos \frac2x - 1} = \lim_{x \to \infty} \frac{\sin^2 \frac{1}{2x}}{\sin^2 \frac{1}{x}} =\lim_{x \to \infty} \frac{\sin^2 \frac{1}{2x}}{(\frac{1}{2x})^2}\frac{(\frac{1}{x})^2}{(\sin^2 \frac{1}{x})}\frac{1}{4}=1*1* \frac14=\frac14.$$</p>
<p>note that $$\lim_{x \to \infty}... |
1,382,087 | <p>Problem:</p>
<p>A bag contains $4$ red and $5$ white balls. Balls are drawn from the bag without replacement.</p>
<p>Let $A$ be the event that first ball drawn is white and let $B$ denote the event that the second ball drawn is red. Find </p>
<p>(i) $P(B\mid A)$</p>
<p>(ii) $P(A\mid B)$</p>
<p>My confusion is t... | zoli | 203,663 | <p>Let the event space be</p>
<p>$$\Omega=\{(r,r),(r,w),(w,r),(w,w)\}$$
the corresponding probabilities are
$$\frac{12}{72},\frac{20}{72},\frac{20}{72},\frac{20}{72}.$$</p>
<p>Then</p>
<p>$$Pr(A\cap B)=Pr((w,r))=\frac{20}{72}$$</p>
<p>and</p>
<p>$$Pr(B)=Pr(\{(r,r),(w,r)\})=\frac{12}{72}+\frac{20}{72}=\frac{32}{72}... |
749,473 | <p>I am trying to model the time it takes until a malfunction appears. For example the time a light-bulb will last. I would like the probability that the light-bulb will burn out at a certain moment (given it hadn't bunt yet) to increase as a function of the time ($P(x | X \geq x$) should be monotonic increasing). That... | Clangon | 142,414 | <p>The Weibull distribution seems to satisfy my request.</p>
|
3,634,416 | <p>First of all, English is not my native language, but Chinses is. I tried to spilt the integration interval into 2 pieces: <span class="math-container">$ [0, 1-1/n] $</span> and <span class="math-container">$ [1-1/n, 1] $</span>. In both intervals I use the mean value theorem:
<span class="math-container">$$
\in... | Riemann | 27,899 | <p>Consider limit
<span class="math-container">$$\lim_{n\to\infty}\int_{0}^{1}\frac{x^n}{1+x^{n}}\,dx=0.$$</span>
then your limit
<span class="math-container">$$\lim_{n\to\infty}\int_{0}^{1}\frac{1}{1+x^{n}}\,dx=1.$$</span>
Hint：
<span class="math-container">$$0\leq\frac{x^n}{1+x^{n}}\leq x^n\implies
\lim_{n\to\infty... |
2,629,133 | <p>In keno, the casino picks 20 balls from a set of 80 numbered 1 to 80. Before the draw is over, you are allowed to choose 10 balls. What is the probability that 5 of the balls you choose will be in the 20 balls selected by the casino?</p>
<p>My attempt: The total number of combinations for the 20 balls is $80\choose... | Parcly Taxel | 357,390 | <p>Without loss of generality, assume the casino picks balls 1 to 20. Then for the stated scenario to happen:</p>
<ul>
<li>Five of your picks are within $[1,20]$: $\binom{20}5$ ways</li>
<li>The other five are within $[21,80]$: $\binom{60}5$ ways</li>
</ul>
<p>There are $\binom{80}{10}$ picks altogether, so the proba... |
2,573,458 | <p>Given $n$ prime numbers, $p_1, p_2, p_3,\ldots,p_n$, then $p_1p_2p_3\cdots p_n+1$ is not divisible by any of the primes $p_i, i=1,2,3,\ldots,n.$ I dont understand why. Can somebody give me a hint or an Explanation ? Thanks.</p>
| drhab | 75,923 | <p>If some integer $m$ is divisible by e.g. $13$ then $m+1$ is not.</p>
<p>Now note that $m=p_1p_2\cdots p_n$ is divisible by every $p_i$ with $i\in\{1,\dots,n\}$ and draw conclusions.</p>
|
3,143,084 | <p>If <span class="math-container">$f : \mathbb{R} \to \mathbb{R}$</span>, we can think of the derivative of <span class="math-container">$f$</span> at a point <span class="math-container">$x$</span>, denoted <span class="math-container">$f'(x)$</span>, as giving the slope of a line tangent to the graph of <span class=... | egreg | 62,967 | <p>Assuming you want to know
<span class="math-container">$$
\lim_{h\to0}\frac{(x+h)^{123}-x^{123}}{h}
$$</span>
set <span class="math-container">$n=123$</span>. In other words, compute
<span class="math-container">$$
\lim_{h\to0}\frac{(x+h)^{n}-x^{n}}{h}
$$</span>
for <em>any</em> integer value of <span class="math-co... |
1,396,322 | <p>For example I have eight kids,</p>
<pre><code>A,B,C,D,E,F,G,H
</code></pre>
<p>If I ask them to go into groups of two, their choices are</p>
<pre><code>A->B
B->C
C->B
D->B
E->A
F->A
G->H
H->C
</code></pre>
<p>How to make sure they get their choices as much as possible?</p>
<p>Or similarl... | tommy | 261,593 | <p>Just expand $\sin(x)$ in a power series and you are done:</p>
<p>$$
\lim_{x\rightarrow 0} \frac{x-\sin(x)}{x^3} =
\lim_{x\rightarrow 0} \frac{x-(x-x^3/3!+\mathcal{O}(x^5))}{x^3}
=\lim_{x\rightarrow 0} (1/3!+\mathcal{O}(x^2))
=1/6
$$</p>
|
240,741 | <p>I'm trying to include the legends inside the frame of the plot like this</p>
<p><a href="https://i.stack.imgur.com/7K5aa.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/7K5aa.jpg" alt="hehe" /></a></p>
<p>Here is my Attempt:</p>
<pre><code>ListPlot[{{2, 5, 2, 8, 6, 8, 3}, {1, 2, 5, 2, 3, 4, 3}},
PlotMark... | Daniel Huber | 46,318 | <p>Often these graphics commands are a bit obscure and one has to try. Is the following approx. what you are looking for?:</p>
<pre><code>ListPlot[{{2, 5, 2, 8, 6, 8, 3}, {1, 2, 5, 2, 3, 4, 3}},
PlotMarkers -> {"\[SixPointedStar]", 15}, Joined -> True,
PlotStyle -> {Orange, Green},
PlotLegends ... |
264,770 | <p>If we have a vector in $\mathbb{R}^3$ (or any Euclidian space I suppose), say $v = (-3,-6,-9)$, then:</p>
<ol>
<li>May I always "factor" out a constant from a vector, as in this example like $(-3,-6,-9) = -3(1,2,3) \implies (1,2,3)$ or does the constant always go along with the vector?</li>
<li>If yes on question 1... | Thomas Andrews | 7,933 | <p>As a rule, this all falls from the distributive rule. If $v=(ax,ay,az)$, then $$\begin{align}||v|| &= \sqrt{(ax)^2 + (ay)^2+(az)^2} = \sqrt{a^2(x^2+y^2+z^2)}\\
&=\sqrt{a^2}\sqrt{x^2+y^2+z^2}\end{align}$$</p>
<p>And $\sqrt{a^2}=|a|$</p>
<p>And yes, ultimately, this is all because we want distances to be pos... |
2,332,277 | <p>First of all, note that $\frac{n^{n+1}}{(n+1)^n} \sim \frac{n}{e}$. </p>
<p><em>Question</em>: Is there $n>1$ such that $n^{n+1} \equiv 1 \mod (n+1)^n$?</p>
<p>There is an OEIS sequence for $n^{n+1}\mod (n+1)^n$: <a href="https://oeis.org/A176823" rel="nofollow noreferrer">https://oeis.org/A176823</a>. </p>
<... | Mastrem | 253,433 | <p>We first prove that it is impossible when $n\not\equiv 1\pmod 4$.</p>
<p>Notice how:
$$n^{n+1}=1+(n-1)\sum_{k=0}^{n}n^k$$
So, we would have $n^{n+1}\equiv 1\pmod{(n+1)^n}$ if and only if:
$$(n+1)^n\mid (n-1)\sum_{k=0}^{n}n^k$$
And since the RHS won't be equal to $0$, we'd have:
$$(n-1)\sum_{k=0}^{n}n^k\ge(n+1)^n$$
... |
2,293,746 | <p>A function f has derivative for all $x\in \mathbb R$ and the limits of $f$ at $+\infty $, $-\infty$ are equal to $+\infty$ . Is it true that $\lim_{x\to a} \frac {1}{f'(x)} = + \infty $ or $-\infty$ for some $a\in\mathbb R$ ?</p>
<p>Of course function $f' $ has roots , according to Fermat's theorem( $f$ has a tot... | Babis Stergiou | 449,018 | <p>Excuse me for(my poor english and ) comig again, I'm new here and I probably I do something in a wrong way.There was a typo in my first messange and the problem exists yet.</p>
<p>My original question is to find if the function $\frac {1}{f'(x)}$ has a vertical asymptote.</p>
<p>The function $f' $ has the Darbo... |
2,304,318 | <p>Let $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ be two real series, where we have $\lim_{n\rightarrow\infty} \frac{a_n}{b_n} = M >0$. Show that either one of these options happen:</p>
<ol>
<li>Both series converge</li>
<li>Both series diverge</li>
</ol>
<p>I have no clue on how to solve this. Can someo... | zhw. | 228,045 | <p>This is false: Define</p>
<p>$$a_n= \frac{(-1)^n}{n^{1/2}},\,\, b_n = \frac{(-1)^n}{n^{1/2}}+\frac{1}{2n^{3/4}}.$$</p>
<p>Then</p>
<p>$$\frac{a_n}{b_n}= \frac{1}{1+(-1)^n/(2n^{1/4})} \to 1.$$</p>
<p>However $\sum a_n$ converges and $\sum b_n$ diverges.</p>
|
2,134,928 | <p>Let <span class="math-container">$ \ C[0,1] \ $</span> stands for the real vector space of continuous functions <span class="math-container">$ \ [0,1] \to [0,1] \ $</span> on the unit interval with the usual subspace topology from <span class="math-container">$\mathbb{R}$</span>. Let <span class="math-container">$$\... | Behemoth | 414,337 | <p>She answered $10$ questions, so she was expecting $30$ points. Instead, she got only $18$ points. That means that she lost a total of $12$ points. If you take into consideration that an incorrect answer takes $4$ points from your expected total ($3$ for annulment, $1$ for penalty), the amount of incorrect answers is... |
211,803 | <p>I ended up with a differential equation that looks like this:
$$\frac{d^2y}{dx^2} + \frac 1 x \frac{dy}{dx} - \frac{ay}{x^2} + \left(b -\frac c x - e x \right )y = 0.$$
I tried with Mathematica. But could not get the sensible answer. May you help me out how to solve it or give me some references that I can go over... | Robert Israel | 8,508 | <p>I don't know if there are closed form solutions in general. In the case $e=0$, Maple finds a solution using Whittaker M and W functions:
$$y \left( x \right) =c_{{1}}
{{\rm \bf M}\left({\frac {ic}{2\sqrt {b}}},\,\sqrt {a},\,2\,i\sqrt {b}x\right)}
{\frac {1}{\sqrt {x}}}+c_{{2}}
{{\rm \bf W}\left({\frac {ic}{2\sqrt ... |
944,840 | <p>For vectors u, w, and v in a vector space V, I am trying to prove:</p>
<p>If $u + w = v + w$ then $u = v$</p>
<p><strong>without</strong> using the additive inverse and only using the 8 axioms which define a vector space. I am coming up short. I don't see how to do this without assuming that if $u + w = v + w$ the... | Barry Cipra | 86,747 | <p>The additive cancellation law you're trying to prove is <em>equivalent</em> to the additive inverse axiom, which strongly suggests you <em>can't</em> prove it without assuming that axiom.</p>
<p>That is, let's take the standard eight axioms from the Wikipedia page for vector spaces, slightly modified by writing $\O... |
3,733,757 | <p>I'm proving that given a nonempty set <span class="math-container">$I$</span>, and given a filter <span class="math-container">$F$</span>, there exists an ultrafilter <span class="math-container">$D$</span> on <span class="math-container">$I$</span> such that <span class="math-container">$F \subseteq D$</span>. I us... | egreg | 62,967 | <p>Let <span class="math-container">$\mathcal{F}$</span> be a filter on <span class="math-container">$I$</span> and take <span class="math-container">$A\subseteq I$</span> such that <span class="math-container">$A\notin\mathcal{F}$</span> and <span class="math-container">$B=I\setminus A\notin\mathcal{F}$</span>.</p>
<p... |
665,759 | <p>Let $G$ be an open subset of $R$. </p>
<p>If $0\notin G$, then show that $H=\{xy:x,y\in G\}$ is an open subset of $R$.</p>
<p>Now Since $G$ is open , given $x,y\in R$, $\exists ,r_x,r_y$ such that $B(x,r_x)\subset G$
and $B(y,r_y)\subset G$.Now all we need to do is find a radius $r$ given a point $xy$ in $H$.</p>... | scineram | 7,598 | <p>Try $r=\max(|x|\cdot r_y,|y|\cdot r_x)$, it's relatively easy.</p>
<p>The optimal radius is $r=|x|\cdot r_y+|y|\cdot r_x-r_x\cdot r_y$.</p>
<p>I want to expand on optimality. For any open set $G\subset\mathbb{R}\setminus\{0\}$ and $x\in G$ denote
$$r_{x,G}:=dist(x,\mathbb{R}\setminus G)$$
and for $x,y\in G$
$$r_{x... |
3,192,795 | <p>Apparently either I've forgotten some basic rule about integrals (it has been a while since I've taken a basic calc class) or something is wrong with this problem in pearson mylab. </p>
<p>This was the problem:</p>
<p>Evaluate <span class="math-container">$\int_C\frac{x^2}{y^{4/3}}ds$</span> where C is the curve <... | kccu | 255,727 | <p>Your antiderivative is incorrect because you are missing a minus sign. Once you substitute the parametrization you should have
<span class="math-container">$$\int_{-3}^{-1} \sqrt{4t^2+9t^4} \ dt.$$</span>
In order to pull the <span class="math-container">$t^2$</span> out of the square root, we need to make it <span ... |
3,192,795 | <p>Apparently either I've forgotten some basic rule about integrals (it has been a while since I've taken a basic calc class) or something is wrong with this problem in pearson mylab. </p>
<p>This was the problem:</p>
<p>Evaluate <span class="math-container">$\int_C\frac{x^2}{y^{4/3}}ds$</span> where C is the curve <... | hamam_Abdallah | 369,188 | <p><span class="math-container">$$\frac{x^2}{y^{\frac 43}}=\frac{t^4}{t^4}=1$$</span></p>
<p>The integral is
<span class="math-container">$$L=\int_C ds$$</span>
it is also the length of the curve between the left point <span class="math-container">$A=(1,-1) $</span> for <span class="math-container">$t=-1$</span> and ... |
370,212 | <p>Let <span class="math-container">$\mathbb{N}$</span> denote the set of positive integers. For <span class="math-container">$\alpha\in \; ]0,1[\;$</span>, let <span class="math-container">$$\mu(n,\alpha) = \min\big\{|\alpha-\frac{b}{n}|: b\in\mathbb{N}\cup\{0\}\big\}.$$</span> (Note that we could have written <span c... | Alapan Das | 156,029 | <p>It's easy to proof that <span class="math-container">$\alpha$</span> shouldn't be a rational number.</p>
<p>Now, let <span class="math-container">$\frac{1}{n-1}>\alpha>\frac{1}{n}, n>1$</span> and <span class="math-container">$\alpha-\frac{1}{n} < \frac{1}{n-1}-\alpha$</span>.</p>
<p>Then, <span class="m... |
863,860 | <p>I am not particularly well-versed in topology, so I wanted to check with you whether there exists a much simpler argument to prove the following statement or whether there are problems with my proof. The statement also seems to be a very standard result but I could not find a reference in e.g. a book on basic topolo... | Graham Kemp | 135,106 | <p>Tip: convert the square roots to exponent form, and combine exponents before taking the derivative.</p>
<p>$$f(x)=\frac{x^2+4x+3}{\sqrt{x}}$$</p>
<p>$$f(x)=(x^2+4x+3)(x^{-1/2})$$</p>
<p>$$f(x)= x^{3/2}+4x^{1/2}+3x^{-1/2}$$</p>
<p>$$f'(x)=\frac 3 2 x^{1/2}+2x^{-1/2}-\frac 3 2 x^{-3/2}$$</p>
<p>$$f'(x)=\frac {3\s... |
3,542,573 | <blockquote>
<p>Solve the differential equation <span class="math-container">$$y''-6y'+25y=50t^3-36t^2 -63t +18$$</span></p>
</blockquote>
<p>I tried solving the homogeneous equation using <span class="math-container">$y = vt$</span>, but I didn't go anywhere. </p>
| Fred | 380,717 | <p>You are not correct. I am missing several <span class="math-container">$T's$</span>.</p>
<p>Correct is</p>
<p><span class="math-container">$$T(2,2,2)=T(2(1,0,0)+2(0,1,1))=2(1,2,3)+2(2,2,2)=(6,8,10).$$</span></p>
|
1,504,483 | <p>Where did the angle convention (in mathematics) come from?</p>
<p>One would imagine that a clockwise direction would be more 'natural' (given
sundials & the like, also a magnetic compass dial).</p>
<p>Also, given time and direction conventions, one would imagine that the
zero degree line would be vertical.</p>... | Community | -1 | <p>It is perhaps "natural" to adopt these two conventions:</p>
<ol>
<li>The zero angle "should" correspond to the positive <span class="math-container">$x$</span>-axis.</li>
<li>Small but positive angles "should" be in the quadrant where <span class="math-container">$x$</span> and <span class="math-container">$y$</spa... |
3,074,900 | <h2>Problem</h2>
<p>When proving one result in the statistical learning theory course, the instructor uses
<span class="math-container">$$
\mathbb{E}[\mathbb{E}[X\vert Y,Z]\vert Z]=\mathbb{E}[X\vert Z]
$$</span>
but I am not sure why this is true.</p>
<h2>What I Have Done</h2>
<p>I know I could do the following
<spa... | angryavian | 43,949 | <p>This is just a special case of the usual
<span class="math-container">$$\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$$</span>
except all expectations are taken under the conditional distribution given the event <span class="math-container">$Z=z$</span>. If you are still unsure, take your favorite proof of the ab... |
3,296,596 | <p>Ive been asked the following question and I'm not sure how to approach it.</p>
<p>Solve the system</p>
<p><span class="math-container">\begin{cases}
x_1+x_2-5x_3=2 \\
6x_1+7x_2+4x_3=7
\end{cases}</span></p>
<p>The answer is required to be in the form of</p>
<p><span class="math-container">$\begin{bmatrix}x_1\\ x... | Salim | 449,176 | <p>Firstly, we get an augmented matrix that represents the linear system of equations.
<span class="math-container">$$
A = \left[\begin{array}{rrr|r}
1 & 1 & -5 & 2 \\
6 & 7 & 4 & 7 \\
0 & 0 & 0 & 0
\end{array}\right]
$$</span></p>
<p>Then we perform the matrix opera... |
229,558 | <p>When we say that a set $S$ is denumerable, that is, there is a bijection $S \to \omega$, do we mean that there <em>exists</em> such a bijection or do we mean that we have one and are talking about a pair $(S,f)$?</p>
<p>I'm asking because it makes a difference to whether I need choice in some proofs or whether I do... | Asaf Karagila | 622 | <p>A set is countable if there exists a bijection with $\omega$. Much like a set is finite if and only if is has a bijection with a finite ordinal. </p>
<p>In the proof that a countable union of countable sets is countable we indeed <em>choose</em> a bijection with $\omega$, and if we were given such bijection to begi... |
2,586,618 | <p>I'm trying to study for myself a little of Convex Geometry and I have some doubts with respect the proof of the Theorem 1.8.5 of the book Convex Bodies: The Brunn-Minkowski Theory. Before I presented the proof and my doubts, I will put the definitions used in the theorem below.</p>
<p><span class="math-container">$\... | daulomb | 98,075 | <p>Using Green's theorem: $$A=(P, Q)= (y, -x)\Longrightarrow\int_L \mathbf{A} \cdot d\mathbf{r}=\displaystyle\int\int_{S}(Q_x-P_y)dA=-2\displaystyle\int\int_{S}dA,$$
where $ S:\, \frac{x^2}{4} + \frac{y^2}{9} \leq 1$. You can change the variables as $x=2u$ and $y=3v$ in which the Jacobian becomes $6$. Then it reduces ... |
4,180,869 | <p>The ReLU activation function in deep learning is given by <span class="math-container">$\text{ReLU}: \mathbb R\rightarrow \mathbb R, x \mapsto \max\left\{0, x\right\}$</span>. I was asking myself whether this function, which is convex, is also closed. This is the general definition of <strong>closed</strong>:</p>
<p... | Community | -1 | <p>I don't know if I can give you a rigorous proof, but I don't think you need one. I think you got to the same conclusion that the epigraph of the ReLU function is <span class="math-container">$\{ (x, y) \colon x \geq 0, y \leq x\}$</span>, which is closed.</p>
|
4,180,869 | <p>The ReLU activation function in deep learning is given by <span class="math-container">$\text{ReLU}: \mathbb R\rightarrow \mathbb R, x \mapsto \max\left\{0, x\right\}$</span>. I was asking myself whether this function, which is convex, is also closed. This is the general definition of <strong>closed</strong>:</p>
<p... | daw | 136,544 | <p>The function is closed if and only if the epigraph is closed. But the ReLu function is continuous, which implies closedness of epigraph.</p>
<p>The epi graph is the set:
<span class="math-container">$$
\{ (x,\alpha) : \max(0,x) \le \alpha\},
$$</span>
which is the preimage of the closed set <span class="math-contai... |
2,406,587 | <p>Isn't the concept of homomorphism and isomorphism in abstract algebra analogous to functions and invertible functions in set theory respectively? That's one way to quickly grasp the concept into the mind?</p>
| Kajelad | 354,840 | <p>Suppose we have a linear transform from $\mathbb R^n\to\mathbb R^n$ defined by a matrix $A$. This transform maps each vector $\vec v\in\mathbb R^n$ to a new vector $A\vec v\in\mathbb R^n$.</p>
<p>An <em>eigenvector</em> of $A$ is simply a nonzero vector $\vec v$ such that $\vec v$ and $A\vec v$ are parallel. Since ... |
995,489 | <p>This is taken from Trefethen and Bau, 13.3.</p>
<p>Why is there a difference in accuracy between evaluating near 2 the expression $(x-2)^9$ and this expression:</p>
<p>$$x^9 - 18x^8 + 144x^7 -672x^6 + 2016x^5 - 4032x^4 + 5376x^3 - 4608x^2 + 2304x - 512 $$</p>
<p>Where exactly is the problem?</p>
<p>Thanks.</p>
| gammatester | 61,216 | <p><span class="math-container">$(x-2)$</span> is small by definition of <span class="math-container">$x$</span>, <span class="math-container">$(x-2)^9$</span> is even much smaller but can be computed with small relative error.
The single terms of the expanded polynomial are much larger and therefore you will suffer fr... |
322,134 | <p>$$2e^{-x}+e^{5x}$$</p>
<p>Here is what I have tried: $$2e^{-x}+e^{5x}$$
$$\frac{2}{e^x}+e^{5x}$$
$$\left(\frac{2}{e^x}\right)'+(e^{5x})'$$</p>
<p>$$\left(\frac{2}{e^x}\right)' = \frac{-2e^x}{e^{2x}}$$
$$(e^{5x})'=5xe^{5x}$$</p>
<p>So the answer I got was $$\frac{-2e^x}{e^{2x}}+5xe^{5x}$$</p>
<p>I checked my answ... | Ross Millikan | 1,827 | <p>You have an extra $x$ in the second term. $(e^{5x})'=5e^{5x}$ by the chain rule. I suspect the online check might prefer $-2e^{-x}$ for the first term, but your version is equivalent.</p>
|
1,948,730 | <blockquote>
<p>For all odd integers $n$, there exists an integer $k$ such that $n=2k+1$.</p>
</blockquote>
<p>I negated using De Morgan's laws. Let $O(n)$ be "$n$ is odd" and $N(n, k)$ "$2k + 1 = n$", then
$$\neg(\forall n \exists k [O(n) \to N(n,k)])\\
\exists n \neg\exists k [O(n) \to N(n,k)]\\
\exists n \forall ... | JMP | 210,189 | <p>How about:</p>
<blockquote>
<p>For all odd integers $n$, there does <strong>not</strong> exist any integer $k$ such that $n=2k+1$</p>
</blockquote>
|
297,907 | <p>Let consider the ring $\mathbb{Z}_p$ and $\zeta$ be a $p$-th root of unity. Especially $\zeta \not \in \mathbb{Z}_p$.
Denote with $\Phi _p(x)$ the cyclotomical polynomial in $p$. Since $p$ is a prime we know that it has the shape $\Phi _p(x)= 1 + x +x^2 +... +x^{p-1}$.
This gives rise for the quotient ring</p>
<p>$... | Laurent Moret-Bailly | 7,666 | <p>Let $m$ be a maximal ideal of $A:=\mathbb{Z}_p[\zeta]$. Then $m\cap\mathbb{Z}_p=p\mathbb{Z}_p$ because $A$ is a finite $\mathbb{Z}_p$-algebra. So the maximal ideals of $A$ are essentially those of $A/pA\cong\mathbb{F}_p[X]/(\Phi_p\bmod p)$. Since $\Phi_p\equiv(X-1)^{p-1}\pmod p$, we see that $A/pA$ is local with max... |
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