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 |
|---|---|---|---|---|
885,129 | <p>I want to speed up the convergence of a series involving rational expressions the expression is $$\sum _{x=1}^{\infty }\left( -1\right) ^{x}\dfrac {-x^{2}-2x+1} {x^{4}+2x^{2}+1}$$
If I have not misunderstood anything the error in the infinite sum is at most the absolute value of the last neglected term. The formula ... | Claude Leibovici | 82,404 | <p>For the summation $$S=\sum _{n=1}^{\infty }\left( -1\right) ^{n}\dfrac {-n^{2}-2n+1} {n^{4}+2n^{2}+1}$$ a CAS gave as a result $$\frac{1}{8} \left(2 \pi ^2 \text{csch}^2\left(\frac{\pi }{2}\right)+(1-i) \left(\psi
^{(1)}\left(\frac{1}{2}-\frac{i}{2}\right)+i \left((-4+4 i)+\psi
^{(1)}\left(\frac{1}{2}+\frac{i}... |
703,125 | <blockquote>
<p>Let $\{A_\alpha\}$ be a collection of connected subspaces of $X$; let
$A$ be connectted subspace of $X$. Show that if $A\cap A_\alpha \neq
\emptyset$ for all $\alpha$, then $A\cup(\cup A_\alpha)$ is connected.</p>
</blockquote>
<p>I know this theorem:<img src="https://i.stack.imgur.com/6tILz.png"... | dani_s | 119,524 | <p>Note that $A \cup A_{\alpha}$ is connected for all $\alpha$ and therefore $A \cup (\bigcup A_{\alpha}) = \bigcup (A \cup A_{\alpha})$ is connected by your theorem because it is the union of connected sets that have $A$ in common.</p>
|
507,454 | <p>I had a geometry class which was proctored using the Moore method, where the questions were given but not the answers, and the students were the source of all answers in the class. One of the early questions which we never solved is listed in the title.</p>
<p>In this case, use any reasonable definition of "betwee... | Zarrax | 3,035 | <p>Hint: Show $p^2 = 1 \mod 3$ and $p^2 = 1 \mod 8$ for any such $p$.</p>
|
3,739,144 | <p>I have the following proposition proved in my lectures notes, but I think there are a couple of errors and there is one think I don't get:</p>
<p>If <span class="math-container">$p_n$</span> is a Cauchy sequence in a metric space <span class="math-container">$(X,d)$</span>, the set <span class="math-container">$\{p_... | Henno Brandsma | 4,280 | <p>Why not say : suppose <span class="math-container">$(p_n)$</span> is Cauchy. Then apply the definition of Cauchy-ness to <span class="math-container">$\varepsilon=1$</span> and we find <span class="math-container">$N \in \mathbb{N}$</span> such that</p>
<p><span class="math-container">$$\forall n,m \ge N: d(p_n, p_m... |
265 | <p>I'm a mod at Computational Science SE. Sometimes, users ask questions on Computational Science about doing something in Wolfram Alpha. As far as I can tell, Wolfram Alpha uses Mathematica for its math engine. Are those questions on topic here?</p>
<p>Note: I was made aware of <a href="https://mathematica.meta.stack... | rm -rf | 5 | <p>I would consider such questions explicitly <strong>off topic</strong> for this site. This site is for users of <em>Mathematica</em> (mma), not Wolfram|Alpha (W|A) and so any question that does not involve the former is out of this site's scope. I know that W|A runs on mma and it kinda sorta understands mma syntax. H... |
3,117,459 | <p>I am interested in approximating the natural logarithm for implementation in an embedded system. I am aware of the Maclaurin series, but it has the issue of only covering numbers in the range (0; 2).</p>
<p>For my application, however, I need to be able to calculate relatively precise results for numbers in the ran... | Oscar Lanzi | 248,217 | <p>In theory, <a href="https://math.stackexchange.com/a/3227686/248217">a series convergent for all positive arguments exists:</a>:</p>
<p><span class="math-container">$\ln(x)=2\sum_{m=1}^\infty {\dfrac{(\frac{x-1}{x+1})^{2m-1}}{2m-1}}$</span></p>
<p>In practice, Arthur's approach of halving the argument until we get t... |
1,722,692 | <p>I am asked to find</p>
<p>$$\lim_{x \to 0} \frac{\sqrt{1+x \sin(5x)}-\cos(x)}{\sin^2(x)}$$</p>
<p>and I tried not to use L'Hôpital but it didn't seem to work. After using it, same thing: the fractions just gets bigger and bigger.</p>
<p>Am I missing something here?</p>
<p>The answer is $3$</p>
| carmichael561 | 314,708 | <p>I think using Taylor expansions is the best way to find the limit, but here is a more elementary approach: multiply the top and bottom by the conjugate to obtain
$$ \lim_{x\to 0}\frac{\sqrt{1+x\sin(5x)}-\cos(x)}{\sin^2(x)}=\lim_{x\to 0}\frac{(1-\cos^2x)+x\sin(5x)}{\sin^2(x)[\sqrt{1+x\sin(5x)}+\cos(x)]} $$</p>
<p>No... |
4,000,935 | <p>How do you show that <span class="math-container">$7a^2 - 12b^2 = 8c^2$</span> has no integer solutions</p>
<hr />
<p>When <span class="math-container">$x = a/c$</span> and <span class="math-container">$y = b/c$</span> then <span class="math-container">$\gcd(a,b,c) = 1$</span> I believe.</p>
<p>If we use mod5, and n... | player3236 | 435,724 | <p>WLOG, assume that <span class="math-container">$\gcd (a,b,c) = 1$</span> as you did.</p>
<p>Taking modulo <span class="math-container">$3$</span> gives <span class="math-container">$a^2 \equiv 2c^2 \pmod 3$</span>.</p>
<p>As <span class="math-container">$n^2 \equiv 0,1 \pmod 3$</span>, we must have <span class="math... |
1,726,416 | <p>$\displaystyle \sum_{n=1}^∞ (-1)^n\dfrac{1}{n}.\dfrac{1}{2^n}$</p>
<p>Knowing that</p>
<ol>
<li>An alternating harmonic series is always convergent</li>
<li>Riemann series are always convergent when $p>1$</li>
</ol>
<p>Is it safe to say that the product of these two is convergent (as described above)?</p>
| Jean Marie | 305,862 | <p>@Kf-Sansoo @Joanpemo</p>
<p>In fact, in my opinion, the right "product" here is the <strong>dot product</strong> $U.V$ in Hilbert space $\ell^2$, which is meaningful because </p>
<p>$$U=(-1, \dfrac{1}{2},-\dfrac{1}{3},\cdots(-1)^n \dfrac{1}{n},\cdots) \ \ and \ \ V=(1,\dfrac{1}{2},\dfrac{1}{4},\cdots\dfrac{1}{2^n}... |
210,401 | <p>In other words, given a sequence $(s_n)$, how can we tell if there exist irrationals $u>1$ and $v>1$ such that </p>
<p>$$s_n = \lfloor un\rfloor + \lfloor vn\rfloor$$</p>
<p>for every positive integer $n$?</p>
<p>A few thoughts: Graham and Lin (<em>Math. Mag.</em> 51 (1978) 174-176) give a test for $(s_n)... | Gerhard Paseman | 3,206 | <p>This is the closest I can come to a positive test, but I don't know how well it will work for you. It is essentially taking the intersection of possible solution sets.</p>
<p>Let us cut down on symmetry by assuming $u \geq v \gt 1$. (I suppose if $u=v$ that would be a tipoff.) For each positive integer $i$, inte... |
4,290,651 | <p>I understand the standard proof that there exists no surjection <span class="math-container">$f: X \to \mathcal{P}(X)$</span>, but I'm not able to tell whether it deals with the case that <span class="math-container">$X = \emptyset$</span> or whether I need to rule this out separately.</p>
<p>If I want to prove that... | Community | -1 | <ul>
<li><p>I don't see why you would need to make a special case for an injection <span class="math-container">$\emptyset\to \{\emptyset\}$</span>, because the function <span class="math-container">$f:\emptyset\to\{\emptyset\}$</span> such that <span class="math-container">$f(x)=\{x\}$</span> for all <span class="math... |
3,072,995 | <p>The only thing I know with this equation is <span class="math-container">$y=\frac{x^2+1}{x+1}=x+1-\frac{2x}{x+1}$</span>.</p>
<p>Maybe it can be solved by using inequality.</p>
| 5xum | 112,884 | <p>Since <span class="math-container">$$\lim_{x\to-\infty} \frac{x^2+1}{x+1} = -\infty,$$</span></p>
<p>the minimum does not exist.</p>
|
2,370,716 | <p>Give the postive integer $n\ge 2$,and $x_{i}\ge 0$,such $$x_{1}+x_{2}+\cdots+x_{n}=1$$
Find the maximum of the value
$$x^2_{1}+x^2_{2}+\cdots+x^2_{n}+\sqrt{x_{1}x_{2}\cdots x_{n}}$$</p>
<p>I try
$$x_{1}x_{2}\cdots x_{n}\le\left(\dfrac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)^n=\dfrac{1}{n^n}$$</p>
| Lazy Lee | 430,040 | <p>Let $x_1\geq x_2 \geq ... \geq x_n$ without loss of generality. Notice that $$\begin{split}F(x_1,...)=\left(\sum_{i=0}^n x_i\right)^2-\sum_{i=0}^n x_i^2 - \sqrt{\prod_{i=0}^n x_i} &= 2\cdot\sum_{i<j}x_ix_j-\sqrt{\prod_{i=0}^n x_i} \\
&=\left(2\sum_{i>1}x_i\right)\cdot x_1-\sqrt{\prod_{i=2}^n x_i}\cdo... |
3,561,807 | <p>There is this question regarding constrained optimisation. It says, a rectangular parallelepiped has all eight vertices on the ellipsoid <span class="math-container">$x^{2}+3y^{2}+3z^{2}=1$</span>. Using the symmetry about each of the planes, write down the surface area of the parallelepiped and therefore find the m... | J. W. Tanner | 615,567 | <p>Let <span class="math-container">$n=2k+1$</span>. Then <span class="math-container">$n^2-1=(n+1)(n-1)=(2k+2)(2k)=4(k+1)k.$</span></p>
|
3,398,645 | <p>I have a doubt about value of <span class="math-container">$e^{z}$</span> at <span class="math-container">$\infty$</span> in one of my book they are mentioning that as <span class="math-container">$\lim_{z \to \infty} e^z \to \infty $</span></p>
<p>But in another book they are saying it doesn't exist.I am confused... | peawormsworth | 603,214 | <p>I think <span class="math-container">$e^x$</span> goes to infinity, while <span class="math-container">$e^z$</span> or <span class="math-container">$(e^i)^x$</span> spins around the unit circle at the rate of the distance around a unit circle.</p>
<p>The letter 'z' could mean Real Integer or Complex number:</p>
<p... |
3,450,908 | <p>I know how to solve problems, but I don't understand the core theory that sticks these two functions with this special relation. </p>
<p>Could you help me find the core idea that I need to understand in order to be satisfied with my understanding? </p>
| g.kov | 122,782 | <p>Just an illustration.</p>
<p>Geometrically, in 2D,
the curves <span class="math-container">$y=\exp(x)$</span> and <span class="math-container">$y=\ln(x)$</span>
are the mirror images of each other
with respect to line <span class="math-container">$y=x$</span>.</p>
<p><a href="https://i.stack.imgur.com/4CxF3.png" ... |
3,370,521 | <p>I have to prove that the following series converges:
<span class="math-container">$$
\sum_{n=1}^{\infty}\frac{\sin n}{n(\sqrt{n}+\sin n)}
$$</span>
I tried to use Dirichlet's test but I was not really sure whether the denominator is a monotonically decreasing function. If it is than the problem becomes easy.</p>
| Kavi Rama Murthy | 142,385 | <p>Certainly not. Consider <span class="math-container">$\frac 1 {z-1}$</span> on the open unit disk. As <span class="math-container">$z \to -1$</span> this function tends to <span class="math-container">$-\frac 1 2 $</span>. </p>
|
68,817 | <p>I have two questions after reading the Hahn-Banach theorem from Conway's book ( I have googled to know the answer but I have not found any result yet. Also I am not sure that whether my questions have been asked here somewhere on this forum - so please feel free to delete them if they are not appropriate )</p>
<p>H... | M.González | 39,421 | <p>When $X^*$ is strictly convex and $M$ is a subspace of $X$, every norm one $f\in M^*$ admits a unique norm one extension $F\in X^*$, because given two of such extensions $F_1$ and $F_2$, $(F_1+F_2)/2$ is norm one. </p>
<p>In fact this property characterizes $X^*$ strictly convex. See Theorem 7.11 in B.V. Limaye "F... |
2,183,809 | <p>The problem:</p>
<blockquote>
<p>If $p$ is prime and $4p^4+1$ is also prime, what can the value of $p$ be?</p>
</blockquote>
<p>I am sure this is a pretty simple question, but I just can't tackle it. I don't even known how I should begin...</p>
| kingW3 | 130,953 | <p>For any $n$ which is not divisible by $5$ you have that $n^4\equiv 1\pmod{5}$ then you must have $4p^4+1\equiv 4+1\equiv 0\pmod{5}$ but $4p^4+1$ must be prime what does that leave you with?</p>
|
2,317,166 | <p>I just pondered this question and have tried out several methods to solve it(mainly using trigonometry). However, I am not satisfied with my trigonometrical proof and is looking for better proofs.</p>
<p>Give an <em>elegant proof</em> that the diagonals are the longest lines in a square.
It would rather be nice if ... | Archis Welankar | 275,884 | <p>Draw a circle circumscribing the square and with end points at opposite corners . Like $A,C $. This circle has diameter as $AC$ and the diameter is the longest chord of circle but this diameter is also the diagonal of the square. Thus the diagonals are the longest lines in a square.</p>
|
2,317,166 | <p>I just pondered this question and have tried out several methods to solve it(mainly using trigonometry). However, I am not satisfied with my trigonometrical proof and is looking for better proofs.</p>
<p>Give an <em>elegant proof</em> that the diagonals are the longest lines in a square.
It would rather be nice if ... | farruhota | 425,072 | <p>It is sufficient to consider the two cases (since others are repetetive): without loss of generality, take two arbitrary points on two sides such that $0\le x\le y\le a$:</p>
<p><a href="https://i.stack.imgur.com/hUeGi.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/hUeGi.png" alt="enter image de... |
493,600 | <p>I am presented with the following task:</p>
<p>Can you use the chain rule to find the derivatives of $|x|^4$ and $|x^4|$ in $x = 0$? Do the derivatives exist in $x = 0$? I solved the task in a rather straight-forward way, but I am worried that there's more to the task:</p>
<p>First of all, both functions is a vari... | Felix Marin | 85,343 | <p>$$
{{\rm d}\left\vert x\right\vert^{4} \over {\rm d}x}
=
4\left\vert x\right\vert^{3}\,{{\rm d}\left\vert x\right\vert \over {\rm d}x}
=
4\left\vert x\right\vert^{3}\,{\rm sgn}\left(x\right)
=
4\left\vert x\right\vert^{2}
\left\lbrack\vphantom{\Large A}%
\left\vert x\right\vert\,{\rm sgn}\left(x\right)
\right\rbrack... |
246,384 | <p>How can I plot a hexagon inside this cylinder so that it is always in the same direction as the cylinder, no matter what the orientation is? In addition, the hexagon must always be the same size as the cylinder.</p>
<p>For example, start at the origin (10,9,8) and go to (1,2,3)?</p>
<pre><code>Graphics3D [{Cylinder ... | asorlik | 77,707 | <p>Summary of issue: dmHalo and dm in the original post were inner parts of a nested data structure in the format <code>{{i, {{j,k,l},{m,n,o},{...}}}}</code>. The problem is that the export formats--.csv, .dat, .txt.-- regardless of keyword--"Data","Table","List"--would convert the inner l... |
1,272,647 | <p>I guess it's a simple question, but it really escaped my memory.</p>
<p>If $a + b =1$, then how can I call those $a$ and $b$ numbers? </p>
<p>$a$ is not an inversion of $b$, and it's not reciprocal of $b$.. but I'm sure that they do have a 'name'.</p>
| Chris Culter | 87,023 | <p>Grammatically, it would make sense to say that $a$ is the <strong>unit complement</strong> or <strong>unity complement</strong> of $b$. Google attests that this phrase is occasionally used, but I wouldn't call it common.</p>
|
2,798,026 | <blockquote>
<p>There are three vector $a$, $b$, $c$ in three-dimensional real vector space, and the inner product between them $a\cdot a=b\cdot b = a\cdot c= 1, a\cdot b= 0, c\cdot c= 4$. When setting $x = b\cdot c$, answer the following question: when $a, b, c$ are linearly dependent, find all possible values of $... | John Douma | 69,810 | <p>Linear dependence places the vectors in the same plane. This makes the problem easier because we can focus on the angle.</p>
<p>From $c\cdot c=4$ we get that $|c|=2$.</p>
<p>So $a\cdot c=2\cos\theta=1\implies\cos\theta=\frac{1}{2}\implies\theta=60^{\circ}$.</p>
<p>From $b\cdot b=1$ and $a\cdot b=0$ we get that $b... |
355,438 | <p>I want to find the geometric locus of point $M$ such that $|MA|^2 |MB|^2=a^2$ where $|AB|=2a$, Solving algebraic equation is not hard but I can't figure out the shape of this curve. Can anybody help?</p>
| MvG | 35,416 | <p>If you want to toy with these shapes, you can try this:</p>
<ol>
<li>Download and install <a href="http://de.cinderella.de/" rel="nofollow noreferrer">Cinderella</a> (free version should be enough)</li>
<li>Start it and add four free points ($A$ through $D$) to the construction</li>
<li>Press <kbd>Ctrl</kbd>+<kbd>E... |
355,438 | <p>I want to find the geometric locus of point $M$ such that $|MA|^2 |MB|^2=a^2$ where $|AB|=2a$, Solving algebraic equation is not hard but I can't figure out the shape of this curve. Can anybody help?</p>
| Narasimham | 95,860 | <p>If points $A,B$ are fixed and negative distances are ruled out, the loci are quartic curves, <em>Ovals of Cassini.</em> (Incorrectly ) he assumed them at first to be planetary orbit loci.</p>
<p><a href="http://www.2dcurves.com/quartic/quarticca.html#Cassiniellipse" rel="nofollow">Ovals_Cassini1</a></p>
<p><a hre... |
1,354,015 | <blockquote>
<p>If <span class="math-container">$a, b, c, d, e, f, g, h$</span> are positive numbers satisfying <span class="math-container">$\frac{a}{b}<\frac{c}{d}$</span> and <span class="math-container">$\frac{e}{f}<\frac{g}{h}$</span> and <span class="math-container">$b+f>d+h$</span>, then <span class="... | Syed Muhammad Asad | 214,964 | <p>As</p>
<p>$$\frac{a+e}{b+f} < \frac{c+g}{d+h}$$
$$(a+e)(d+h) < (c+g)(b+f)$$
$$ad+eh+ah+ed < cb+fg+cf+gb$$
here put $ah+ed = x$ and $cf+gb = y$. Now we have
$$ad+eh+x < bc+fg+y$$
Now as the given
$$\frac{a}{b}<\frac{c}{d} \implies { ad<bc}$$
and
$$\frac{e}{f}< \frac{g}{h} \implies{eh<fg}$$
so... |
275,308 | <p>Problems with calculating </p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}$$</p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}=\lim_{x\rightarrow0}\frac{\ln(2\cos^{2}(x)-1)}{(2\cos^{2}(x)-1)}\cdot \left(\frac{\sin x}{x}\right)^{-1}\cdot\frac{(2\cos^{2}(x)-1)}{x^{2}}=0$$</p>
<p>Correct answer i... | Pedro | 23,350 | <p>The known limits you might wanna use are, for $x\to 0$</p>
<p>$$\frac{\log(1+x)}x\to 1$$</p>
<p>$$\frac{\sin x }x\to 1$$</p>
<p>With them, you get</p>
<p>$$\begin{align}\lim\limits_{x\to 0}\frac{\log(\cos 2x)}{x\sin x}&=\lim\limits_{x\to 0}\frac{\log(1-2\sin ^2 x)}{-2\sin ^2 x}\frac{-2\sin ^2 x}{x\sin x}\\&a... |
1,785,414 | <p>I am trying to find a closed form for the integral $$I=\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac{\lfloor|\tan x|\rfloor}{|\tan x|}dx$$ So far, my reasoning is thus: write, by symmetry through $x=\pi/2$, $$I=2\sum_{n=1}^{\infty}n\int_{\arctan n}^{\arctan (n+1)}\frac{dx}{|\tan x|}=2\sum_{n=1}^{\infty}n\ln\frac{\sin\a... | David H | 55,051 | <p>$$\begin{align}
I
&=\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\frac{\lfloor\left|\tan{\left(x\right)}\right|\rfloor}{\left|\tan{\left(x\right)}\right|}\,\mathrm{d}x\\
&=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{\lfloor\left|\cot{\left(y\right)}\right|\rfloor}{\left|\cot{\left(y\right)}\right|}\,\mathrm{d}y;~~~\s... |
66,801 | <p>In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have progressed to chapter 3). My intention is to read this book and then ... | Spencer | 4,281 | <p>To my mind, the classical subject is quite different from the modern, evolved form of the subject</p>
<p>I started on the classical side with Yitzhak Katznelson's <em>An Introduction to Harmonic Analysis</em>: This is in the classical camp: Lots on Fourier Series. Very clear; very nice proofs. You will learn lots o... |
66,801 | <p>In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have progressed to chapter 3). My intention is to read this book and then ... | Martin Peters | 60,435 | <p>For the link to applications you might try Palle Jorgensen´s: <em>Analysis and Probability. Wavelets, Signals, Fractals.</em> As a special feature, this book contains a dictionary of the use of technical terms in different disciplines.</p>
|
1,025,117 | <p>Let $V$ be finite dim $K-$vector space. If w.r.t. any basis of $V$, the matrix of $f$ is a diagonal matrix, then I need to show that $f=\lambda Id$ for some $\lambda\in K$. </p>
<p>I am trying a simple approach: to show that $(f-\lambda Id)(e_i)=0$ where $(e_1,...,e_2)$ is a basis of $V$. Let the diagonal matrix b... | DeepSea | 101,504 | <p>Let $u = \dfrac{c}{x}$, then: </p>
<p>Note that: $x\log\left(\dfrac{x+c}{x-c}\right) = c\dfrac{\log\left(\dfrac{1+u}{1-u}\right)}{u} = c\left(\dfrac{\log(1+u)}{u} - \dfrac{\log(1-u)}{u}\right) \to c(1-(-1)) = 2c$ since $x \to \infty \iff u \to 0$.</p>
|
2,186,743 | <p>I think this is a very basic question but somehow I am unable to understand the answer.</p>
<hr>
<blockquote>
<p>Find the number of zeroes of $f(x) = x^3 + x + 1$.</p>
</blockquote>
<hr>
<p>Answer in the book : </p>
<p>$f^\prime (x) = 3x^2 + 1$ since $3x^2 + 1 \ge 0$ for all $x \in \mathbb{R}$ therefore $f^\p... | Abdullah al allah | 416,261 | <p>The derivative is always positive, so the function is monotonically increasing, hence it will have one root at most if the minimum value is less than 0.</p>
|
3,298,311 | <p>How can I prove (by definition) that, if <span class="math-container">$a, b \in \mathbb{R}$</span> and <span class="math-container">$a<b$</span>, then <span class="math-container">$[a, b]$</span> is equal to the set of accumulation (limit) points?</p>
<p>Let <span class="math-container">$(E, d)$</span> a metric ... | Mike Pierce | 167,197 | <p>If <span class="math-container">$x \in [a,b]$</span>, the <span class="math-container">$x \in B_\varepsilon(x) \cap [a,b]$</span> for all <span class="math-container">$\varepsilon >0$</span>. If <span class="math-container">$x \notin [a,b]$</span> take <span class="math-container">$\varepsilon = \frac{1}{2}\min\{... |
445,127 | <p>I need to prove this limit:</p>
<blockquote>
<p>Given $f:(-1,1) \to \mathbb{R}\,$ and $\,f(x)>0,\,$ if $\,\lim_{x\to 0} \left(f(x) + \dfrac{1}{f(x)}\right) = 2,\,$ then $\,\lim_{x\to 0} f(x) = 1$.</p>
</blockquote>
| zuggg | 84,273 | <p>Assume $f(x)$ does not converge towards $1$ when $x\to 0$. That means there exists $\epsilon>0$ and a sequence $(x_n)_{n\in\mathbb{N}}$ such that $x_n\to 0$, and $|f(x_n)-1|>\epsilon$ for all $n$, so either $f(x_n)>1+\epsilon$ or $f(x_n)<1-\epsilon$.</p>
<p>Now, notice that the function $g:x\mapsto x+1/... |
445,127 | <p>I need to prove this limit:</p>
<blockquote>
<p>Given $f:(-1,1) \to \mathbb{R}\,$ and $\,f(x)>0,\,$ if $\,\lim_{x\to 0} \left(f(x) + \dfrac{1}{f(x)}\right) = 2,\,$ then $\,\lim_{x\to 0} f(x) = 1$.</p>
</blockquote>
| Hagen von Eitzen | 39,174 | <p>Let $g(x)=\max\{f(x),\frac1{f(x)}\}$ and note that $$\begin{align}h\colon [1,\infty)&\to[2,\infty)\\x&\mapsto x+\frac1x\end{align}$$ is a continous bijection with continuous inverse $$\begin{align}h^{-1}\colon [2,\infty)&\to[1,\infty)\\y&\mapsto \left(\frac{\sqrt{y-2}+\sqrt{y+2}}2\right)^2\end{align}... |
445,127 | <p>I need to prove this limit:</p>
<blockquote>
<p>Given $f:(-1,1) \to \mathbb{R}\,$ and $\,f(x)>0,\,$ if $\,\lim_{x\to 0} \left(f(x) + \dfrac{1}{f(x)}\right) = 2,\,$ then $\,\lim_{x\to 0} f(x) = 1$.</p>
</blockquote>
| N. S. | 9,176 | <p><strong>Hint</strong> Prove first that there exists some <span class="math-container">$\delta$</span> such that <span class="math-container">$f$</span> is <strong>positive and bounded</strong> on <span class="math-container">$(x- \delta, x+ \delta)$</span>.</p>
<p><strong>Hint 2:</strong>
<span class="math-containe... |
1,690,092 | <p>$a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + ... + \frac{1}{\sqrt{n^2+2n-1}}$</p>
<p>and I need to check whether this sequence converges to a limit without finding the limit itself. I think about using the squeeze theorem that converges to something (I suspect '$1$').</p>
<p>But I wrote $a_{n+1}$ ... | GoodDeeds | 307,825 | <p>$$\frac{1}{\sqrt{n^2+2n-1}}+\frac{1}{\sqrt{n^2+2n-1}}\cdots+\frac{1}{\sqrt{n^2+2n-1}}\le\frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + \cdots + \frac{1}{\sqrt{n^2+2n-1}}\le\frac{1}{\sqrt{n^2+n}}+\frac{1}{\sqrt{n^2+n}}\cdots+\frac{1}{\sqrt{n^2+n}}$$
$$\frac{n}{\sqrt{n^2+2n-1}}\le\frac{1}{\sqrt{n^2+n}} + \frac{1... |
626,928 | <p>I took linear algebra course this semester (as you've probably noticed looking at my previously asked questions!). We had a session on preconditioning, what are they good for and how to construct them for matrices with special properties. It was a really short introduction for such an important research topic, so I'... | Brian Rushton | 51,970 | <ul>
<li>Here's a famous survey paper that may be too long: <a href="http://www.mathcs.emory.edu/~benzi/Web_papers/survey.pdf" rel="nofollow">Preconditioning
Techniques for Large Linear Systems: A Survey by Michele
Benzi</a></li>
<li>Then there's this shorter introduction that is heavy on computer
programming and matla... |
390,771 | <p>$$\int\frac{dx}{(1+x^\frac{1}{4})x^\frac{1}{2}}$$
This is my work:
$$u^4=x$$
$$4u^3=dx$$
$$\int\frac{4u^3du}{(1+u)u^2}=\int\frac{4u^3du}{(1+u)u^2}=-4(1+x^\frac{1}{4})^{-1}+2(1+x^\frac{1}{4})^{-2}+C$$
But <a href="http://www.wolframalpha.com/input/?i=integrate%201/%28%281%2bx%5E%281/4%29%29x%5E%281/2%29%29" rel="nofo... | Norbert | 19,538 | <p>Check this calculations. They give the same result as wolfram alpha
$$
\int\frac{4u}{1+u}du=
4\int\left(1-\frac{1}{u+1}\right)du=
4\int du-4\int\frac{1}{u+1}du=
4u-4\ln(1+u)=\\
4x^{1/4}-4\ln(1+x^{1/4})
$$</p>
|
390,771 | <p>$$\int\frac{dx}{(1+x^\frac{1}{4})x^\frac{1}{2}}$$
This is my work:
$$u^4=x$$
$$4u^3=dx$$
$$\int\frac{4u^3du}{(1+u)u^2}=\int\frac{4u^3du}{(1+u)u^2}=-4(1+x^\frac{1}{4})^{-1}+2(1+x^\frac{1}{4})^{-2}+C$$
But <a href="http://www.wolframalpha.com/input/?i=integrate%201/%28%281%2bx%5E%281/4%29%29x%5E%281/2%29%29" rel="nofo... | amWhy | 9,003 | <p>You substitutions are just fine. First, we cancel the common factor of $u^2$ in the numerator and denominator.
$$\int\frac{4u^3du}{(1+u)u^2}=\int\frac{4u\,du}{1+u}$$</p>
<p>Dividing the numerator by the denominator gives us $$\int\frac{4u}{1+u}\,du = \int \left(4 - \frac 4{1+u} \right) \,du = 4\int \,du - 4\int \fr... |
2,333,083 | <p>I want to know about when the infinite product form of the Riemann Zeta Function converges.</p>
<p>It is easy to show that $$\sum_{n=1}^\infty \frac 1{n^x}$$ converges for $\Bbb R(x)>1$.</p>
<p>When does $$\prod_{p=primes} \frac {p^x}{p^x-1}$$ converge? And how do you show that?</p>
| reuns | 276,986 | <p>Let $$A_k = \{ n \in \mathbb{N}, \text{ largest prime factor of } n \le k\}$$
Using the fundamental theorem of arithmetic, or by induction
$$\prod_{p \in \mathcal{P},p \le k} (1+p^{-s}+(p^2)^{-s}+(p^3)^{-s}+\ldots) = \sum_{n \in A_k} n^{-s}$$</p>
<p>For $s > 1$ the sequence $\{ n^{-s}\}_{n\in \mathbb{N}}$ is non... |
2,676,200 | <blockquote>
<p>A hyperbola has equation $\frac{x^2}{4}-\frac{y^2}{16}=1$. Show that every other line parallel to this asymptote, $y=2x$, intersects the hyperbola exactly once.</p>
</blockquote>
<p><a href="https://i.stack.imgur.com/gFB6v.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/gFB6v.png" ... | Álvaro Serrano | 535,487 | <p>You don't have a quadratic equation, $c^2+4cx+16=0$ is a linear equation with the single solution $x=-\frac{16+c^2}{4c}$, so there is only one intersection point when $c \ne 0$ and none when $c = 0$.</p>
|
1,893,609 | <p>I am trying to show that $A=\{(x,y) \in \Bbb{R} \mid -1 < x < 1, -1< y < 1 \}$ is an open set algebraically. </p>
<p>Let $a_0 = (x_o,y_o) \in A$. Suppose that $r = \min\{1-|x_o|, 1-|y_o|\}$ then choose $a = (x,y) \in D_r(a_0)$. Then</p>
<p>Edit: I am looking for the proof of the algebraic implication t... | Mick | 42,351 | <p>Four points constitute a quadrilateral. If these four points lie on the same circle, the quadrilateral thus formed is called cyclic quadrilateral. A cyclic quadrilateral can be but need not be a rectangle. A cyclic quadrilateral has the properties of the sum each pair of interior opposite angles is $180^0$.</p>
<p>... |
11,629 | <p>The basic logic course in school gives the impression that logic has both the syntax and the semantics aspects. Recently, I wonder whether the syntax part still plays an essential role in the current studies. Below are some of my observations, I hope the idea from the community can make them more complete.</p>
<p>M... | Charles Stewart | 3,154 | <p>We don't know how to abstract away from syntax in proof theory. If we say there are three main branches in proof theory:</p>
<ol>
<li>Axiomatics seem to be necessarily syntactic: formulae are what it is about;</li>
<li>Relative consistency & ordinal analysis, which are ultimately about characterising provabili... |
11,629 | <p>The basic logic course in school gives the impression that logic has both the syntax and the semantics aspects. Recently, I wonder whether the syntax part still plays an essential role in the current studies. Below are some of my observations, I hope the idea from the community can make them more complete.</p>
<p>M... | Justin Hilburn | 333 | <p>To elaborate slightly on Neel's post: the fact that not all proofs in intuitionist and substructural logics are identified is a strength and not a weakness when viewed through the lens of the Curry-Howard isomorphism which shows that logic has computational content.</p>
<p>Basically the Curry-Howard isomorphism sta... |
1,952,562 | <p>Find an equation of the tangent line to the curve $y = sin(3x) + sin^2 (3x)$ given the point (0,0). Answer is $y = 3x$, but please explain solution steps. </p>
| Emilio Novati | 187,568 | <p>Hint:</p>
<p>Do you know that the slope of the tangent line at a point of the graph of a function is the derivative of the function at this point?</p>
<p>So, for $y = \sin(3x) + \sin^2 (3x)$ find the derivative $y'$ ( can you do?), then evaluate this derivative for $x=0$ </p>
<p>Now the line has equation $y=mx$... |
628,236 | <p>I am trying to give a name to this axiom in a definition: </p>
<p>$(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$</p>
<p>(for all $X, Y, R, S$) where $\sqcup$ is the join of a lattice and $\bullet$ is some binary operation. It feels related to monotonicity/distributivity but I don't k... | Community | -1 | <p>Suppose that $Z^2-YZ-Y^2+X^2+2XY$ is reducible. Since this is a degree two polynomial in $Z$ with coefficients in $\Bbb C[X,Y]$ it splits into a product of two polynomials of degree one in $Z$, that is, $Z^2-YZ-Y^2+X^2+2XY=(Z+f(X,Y))(Z+g(X,Y))$. We get $f(X,Y)+g(X,Y)=-Y$ and $f(X,Y)g(X,Y)=-Y^2+X^2+2XY$. Now plug in ... |
31,261 | <p>Given a 3-manifold $M$, one can define the Kauffman bracket skein module $K_t(M)$ as the $C$-vector space with basis "links (including the empty link) in $M$ up to ambient isotopy," modulo the skein relations, which can be found in the second paragraph of section two of <a href="http://arxiv.org/abs/math/0402102" re... | Bruce Westbury | 3,992 | <p>This is not an answer, more like one comment and one suggestion for an approach to this problem.</p>
<p>The comment is that this looks like a 4-dimensional TQFT. You have an algebra associated to a surface, a module associated to a 3-manifold, and a vector associated to a 4-manifold. The reason it is not usually pr... |
2,164,946 | <p>I'm working on arc length calculation and area of surface of revolution in calculus and I'm really quite stuck on the process of how to do this. Here is a particular problem that I'm struggling with:</p>
<blockquote>
<p>Find the surface area of the surface of revolution generated by revolving the graph $$y=x^3; \... | Mansour Alkatheeri | 443,886 | <p>$X^3$ should be placed inside the integration not outside</p>
|
195,006 | <p>I am not very familiar with mathematical proofs, or the notation involved, so if it is possible to explain in 8th grade English (or thereabouts), I would really appreciate it.</p>
<p>Since I may even be using incorrect terminology, I'll try to explain what the terms I'm using mean in my mind. Please correct my term... | Brian M. Scott | 12,042 | <p>If I understand you correctly, what you mean by a <em>sequentially infinite set</em> is probably a countably infinite set with a specific ordering attached that makes it possible to specify a first element, a second element, and so on. In other words, you have what amounts to an infinite list:</p>
<p>$$a_1,a_2,a_3,... |
1,147,373 | <p>I need to prove that
<span class="math-container">$$I = \int^{\infty}_{-\infty}u(x,y) \,dy$$</span>
is independent of <span class="math-container">$x$</span> and find its value, where
<span class="math-container">$$u(x,y) = \frac{1}{2\pi}\exp\left(+x^2/2-y^2/2\right)K_0\left(\sqrt{(x-y)^2+(-x^2/2+y^2/2)^2}\right)$$<... | Upax | 157,068 | <p>I think there is a typo somewhere. I tell you why. First of allI write your equation as folllow
<span class="math-container">\begin{equation}
u(x,y)=g(x,y) K_0(\rho(x,y))
\end{equation}</span>
where
<span class="math-container">\begin{equation}
g(x,y) =\frac{1}{2\pi}\exp\left(+x^2/2-y^2/2\right)
\end{equation}</span... |
3,513,581 | <p>I'm referring to Sakasegawa's forumla for calculating average line length in a queuing system.</p>
<p><a href="https://i.stack.imgur.com/Ydg3Y.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ydg3Y.png" alt="enter image description here"></a></p>
<p>I don't understand the result intuitively. For... | shortmanikos | 729,993 | <p>Disclaimer: I first heard of queueing theory today :)</p>
<p>I think the key is that 8 customers per hour do not arrive at a steady rate of 1 every 7.5 minutes but are randomly distributed. Taking an extreme example say you have 16 people appear at the start of the day. Nobody else arrives for the next two hours. <... |
3,201,996 | <p>I have an 8-digit number and you have an 8-digit number - I want to see if our numbers are the same without either of us passing the other our actual number. Hashing the numbers is the obvious solution. However, if you send me your hashed number and I do not have it - it is very easy to hash all the permutations of ... | kelalaka | 338,051 | <p>In Cryptography, There are two interesting problems;</p>
<ol>
<li><a href="https://en.wikipedia.org/wiki/Yao%27s_Millionaires%27_Problem" rel="nofollow noreferrer">Yao's Millionaires Problem</a>: In which two millionaires want to know who is the reacher without revealing their riches to the other.</li>
<li><a href=... |
2,547,888 | <p>A $\subset$ B, A and B closed then B-A is open</p>
<p>How can I prove this statement? Is that correct? I need it because is used to prove that a closed subset of a compact space is compact. </p>
<p>Imagine B=[0,4] and A=[1,2] contained in A, then B-A is neither open or closed, what am I doing wrong??</p>
<p>PS: ... | hmakholm left over Monica | 14,366 | <p>What you're missing is that for "closed subset of a compact space is compact", $A$ is not only compact and closed: $A$ <em>is the entire space</em> and therefore <em>open</em>. Thus $A\setminus B = A\cap \overline B$ is open too.</p>
|
176,987 | <p>Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ whose entries are also chosen independently and uniformly from $\{0,1\}$. Let $N = Mv$ where multiplication is performed... | john mangual | 1,358 | <p>This reminds me a bit of the game of <a href="https://en.wikipedia.org/wiki/Mastermind_(board_game)" rel="nofollow noreferrer">mastermind</a>.</p>
<p><img src="https://cimsec.org/wp-content/uploads/2013/04/mastermind13.jpg" width="200"></p>
<hr>
<p>Since your matrix is <a href="https://en.wikipedia.org/wiki/Circu... |
2,018,071 | <p>The matrix is banded and symmetric, and is positive definite. It's very similar to a Laplacian matrix, though it's slightly modified, signs are flipped and every value is divided by the same constant. I need to find the smallest two eigenvalues in MATLAB, without using the eig() function. </p>
<p>Since I know it's ... | Max | 130,322 | <p>i might be wrong, because i think my idea sounds really naive: let $\lambda_{M}$ be the largest eigenvalue of $M$. (It should be possible to get a good approximation for the greatest eigenvalue of any symmetric positive definite matrix at least by an iterative method) Let $C\geq \lambda_M$ be any constant (to compen... |
2,018,071 | <p>The matrix is banded and symmetric, and is positive definite. It's very similar to a Laplacian matrix, though it's slightly modified, signs are flipped and every value is divided by the same constant. I need to find the smallest two eigenvalues in MATLAB, without using the eig() function. </p>
<p>Since I know it's ... | pixel | 27,824 | <p>Matlab has <code>eigs</code> to give the largest or smallest few eigenvalues of a matrix. See the <a href="http://www.mathworks.com/help/matlab/ref/eigs.html" rel="nofollow noreferrer">documentation</a>. The <code>sm</code> option takes the matrix inverse, but the <code>sa</code> option does not, which might be more... |
1,590,262 | <p>Let $ABC$ be of triangle with $\angle BAC = 60^\circ$
. Let $P$ be a point in its interior so that $PA=1, PB=2$ and
$PC=3$. Find the maximum area of triangle $ABC$.</p>
<p>I took reflection of point $P$ about the three sides of triangle and joined them to vertices of triangle. Thus I got a hexagon having area doubl... | nikola | 213,926 | <p>Let E be the point outside triangle such that AEPC is a parallelogram , let F be point outside the triangle such that CPBF is a parallelogram, and let D be a point outside ABC such that APBC is paralellogram. Now, you have hexagon AECFBD with sides EC=1,CF=2,FB=3,BD=1, AD=2, EA=3, and draw line EB. Then, ADBE and EB... |
1,977,577 | <p>if a function is lebesgue integrable, does it imply that it is measurable?
(without any other assumption)</p>
<p>The reason why I ask this is because royden, in his book, kind of imply about a measurable function when assuming the function to be lebesgue integrable</p>
| Adam | 82,101 | <p>By definition a function $f$ is called Lebesgue integrable if $f$ is measurable and $$\int |f(x)| \, \mu(dx) < \infty.$$</p>
<p>This has to be included in the definition because otherwise the term $$ \int |f(x)| \, \mu(dx)$$ is not defined.</p>
|
3,283,600 | <p>"Find all the complex roots of the following polynomials</p>
<p>A) <span class="math-container">$S(x)=135x^4 -324x^3 +234x^2 -68x+7$</span>, knowing that all its real roots belong to the interval <span class="math-container">$(0.25;1.75)$</span></p>
<p>B)<span class="math-container">$M(x)=(x^3 -1+i)(5x^3 +27x^2 -2... | Lubin | 17,760 | <p>I’ll deal with (B) only. You have probably seen that <span class="math-container">$\frac35$</span> is a root, so that <span class="math-container">$x-\frac35$</span> is a factor, and when you divide <span class="math-container">$5x^3 +27x^2-28x+6$</span> by that, you get a <span class="math-container">$\Bbb Q$</spa... |
1,303,183 | <p>If I have a vector space $V$ ( of dimension $n$ ) over real numbers such that $\{v_1,v_2...v_n\}$ is the basis for the space ( not orthogonal ). Then I can write any vector $l$ in this space as $l=\sum_i\alpha_iv_i$. Here $\alpha_1,\alpha_2...\alpha_n$ are the coefficients that define the vector $l$ according to th... | muaddib | 242,554 | <p>They cannot. Suppose so. Then $\sum \alpha_i v_i = \sum \beta_i v_i$. This implies:
$$0 = \sum (\alpha_i - \beta_i) v_i$$
which violates the basis being a set of linearly independent vectors.</p>
|
258,215 | <p>How can we show that if $f:V\to V$
Then for each $m\in \mathbb {N}$ $$\operatorname{im}(f^{m+1})\subset \operatorname{im}(f^m)$$
Please help,I am stuck on this.</p>
| Chris Eagle | 5,203 | <p>No, of course not. For example, let $f$ be a constant function, $f(x)=c$. Then for any $B\subset Y$, $f^{-1}(B)$ is either $X$ (if $c \in B$) or $\varnothing$ (if $c \not \in B$). So if $X$ has more than one element, then $f^{-1}$ is not surjective.</p>
|
1,549,490 | <p>$f(x) = 3x - \frac{1}{x^2}$</p>
<p>I am finding this problem to be very tricky:</p>
<p><a href="https://i.stack.imgur.com/7RjYG.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7RjYG.jpg" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/jwUHp.jpg" rel="nofollow n... | Dr. Sonnhard Graubner | 175,066 | <p>we have $$\frac{f(x+h)-f(x)}{h}=\frac{3(x+h)-\frac{1}{(x+h)^2}-3x+\frac{1}{x^2}}{h}$$ for the numerator of the right-hand side we obtain
$$\frac{h \left(3 h^2 x^2+6 h x^3+h+3 x^4+2 x\right)}{x^2 (h+x)^2}$$ thus we have $$\frac{f(x+h)-f(x)}{h}=\frac{3h^2x^2+6hx^3+h+3x^4+2x}{x^2(h+x)^2}$$
now you can calculate the lim... |
905,672 | <p>Let $S$ be a subset of a group $G$ that contains the identity element $1$ and such that the left cosets $aS$ with $a$ in $G$, partition $G$.Prove that $S$ a is a subgroup of $G$.</p>
<p>My try:</p>
<p>For $h$ in $S$, If I show that $hS=S$, then that would imply that $S$ is closed. </p>
<p>Now $hS$ is a partiton o... | Murtuza Vadharia | 132,945 | <p>It was difficult to follow your argument, but I think you got it right.
My try:
We will prove that S is closed under multiplication. Let a,b∈S be arbitrary. Since ab∈aS, it will be sufficient to prove that aS=S. Since e∈S, we have a=ae∈aS. By assumption, we also have a∈S=eS. Since aS and eS are both cells of the par... |
2,494,069 | <p>All the calculators I've tried overflow at 2^1024. I'm not sure how to go about calculating this number and potentially even larger ones.</p>
| James Garrett | 457,432 | <p>Just use the property $\ln(2^{2048})=2048\cdot \ln(2)$. Hence the answer is approximately $2048\cdot 0,693=1419.264$.</p>
|
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| Joffan | 206,402 | <p>Take a circle with circumference $2$, and place the first point arbitrarily. Then the second point which will fall a distance $a$ away from that first point, where $a$ is somewhere between $0$ and $1$, with equal probability.</p>
<p>Then in order for the triangle to contain the centre of the circle, the third point... |
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| Isaak | 560,931 | <p>Okay so not a math whiz or anything, so don't expect an elegant answer, but I did this question earlier today because I saw it online, and this is basically what I got:</p>
<p>You're placing 3 points (A, B and C) randomly around the circumference of a circle. What's the probability that a triangle drawn between tho... |
2,701,658 | <p>If <span class="math-container">$1^2+2^2+3^2 + ... + 10^2=385$</span>, then what is the value of <span class="math-container">$2^2+4^2+6^2 + ... + 20^2$</span>?</p>
<p>Options are <span class="math-container">$770$</span>, <span class="math-container">$1155$</span>, <span class="math-container">$1540$</span>, <span... | user061703 | 515,578 | <p>$1^2+2^2+3^2+...+10^2=385$</p>
<p>$\Rightarrow 4\times(1^2+2^2+3^2+...+10^2)=1540$</p>
<p>$\Rightarrow 1^2\times 4+2^2\times 4+3^2 \times 4+...+10^2 \times 4=1540$</p>
<p>$\Rightarrow 1^2\times 2^2+2^2 \times 2^2+3^2\times 2^2+...+10^2\times 2^2=1540$</p>
<p>$\Rightarrow (1\times2)^2+(2\times2)^2+(3\times2)^2+(4... |
3,080,664 | <p>Hi I've almost completed a maths question I am stuck on I just can't seem to get to the final result. The question is:</p>
<p>Find the Maclaurin series of <span class="math-container">$g(x) = \cos(\ln(x+1))$</span> up to order 3.</p>
<p>I have used the formulas which I won't type out as I'm not great with Mathjax ... | Keen-ameteur | 421,273 | <p><span class="math-container">$f(x)=o(x^k)$</span> if:</p>
<p><span class="math-container">$\underset{x\rightarrow 0}{\lim} \frac{f(x)}{x^k}=0$</span></p>
<p>Since smaller powers of <span class="math-container">$x$</span> matter more when tending to <span class="math-container">$0$</span>, it essentially means that... |
4,184,248 | <p>I'm trying to prove an inequality from an old book named "Algebra and elementary functions" by Kochetkov (ru - "Алгебра и элементарные функции" Е. С. Кочетков). This is an old book from 1970 (URSS era). The book is not copyrighted anymore, and it is freely available over the internet.</p>
<p>The ... | cansomeonehelpmeout | 413,677 | <p>Assume there are 3 distinct roots <span class="math-container">$r_1,r_2,r_3$</span>. Then we have the system of equations <span class="math-container">$$lr_1^2+mr_1+n=0\\lr_2^2+mr_2+n=0\\lr_2^2+mr_2+n=0\\$$</span> as <span class="math-container">$r_1\neq r_2\neq r_3$</span> we can add and subtract lines to get <span... |
1,220,002 | <p>I am performing the ratio test on series and I come across many situations where I have any of those 3 in the numerator and denominator and I was wondering what I could cancel as n approaches infinity.</p>
| Elaqqad | 204,937 | <p>We have the following orders we have $n!<<n^n$ and $(n+1)^n\sim e n^n$:</p>
<ul>
<li>When we are dealing with fractions with theses terms the best approsimation for $n!$ is given by Stirling's formula:
$$n\sim \left (\frac{n}{e} \right)^n\sqrt{2\pi n} $$</li>
<li>Note also that:
$$\left (1+\frac{x}{n}\right)^... |
1,220,002 | <p>I am performing the ratio test on series and I come across many situations where I have any of those 3 in the numerator and denominator and I was wondering what I could cancel as n approaches infinity.</p>
| hunter | 108,129 | <p>No, neither of these are true. First,
$$
\lim_{n \to \infty} \frac{(n+1)^n}{n^n} = e.
$$
This is because
$$
\frac{(n+1)^n}{n^n} = \left(\frac{n+1}{n}\right)^n = \left(1 + \frac{1}{n}\right)^n
$$
The limit of the last expression is the definition of $e$.</p>
<p>On the other hand
$$
\lim_{n \to \infty} \frac{n!}{n^n}... |
1,176,381 | <p>I have many sets containing three values like $\{1, -2, 5\}$.
I am want to write in mathematical form to filter set where exist one value with different sign. (and for sure, none of them should be zero)</p>
<p>I am not sure about tag. (please correct the tag if it is not correct)/ </p>
| Community | -1 | <p>How about ($\mathcal M$ is your family of sets) $\mathcal M \cap \left(F_1 \cup F_2 \cup F_3\right)$, where:
$$F_1 = \{\{a,b,c\} : a,b \in \mathbb R_+, c \in \mathbb R_-\} \\
F_2 = \{\{b,c,a\} : a,b \in \mathbb R_+, c \in \mathbb R_-\} \\
F_3 = \{\{c,a,b\} : a,b \in \mathbb R_+, c \in \mathbb R_-\}. \\
$$</p>
|
2,472,313 | <blockquote>
<p>Suppose $x$ and $y$ are real numbers and $x^2+9y^2-4x+6y+4=0$. Then find the maximum value of $\left(\frac{4x-9y}{2}\right)$</p>
</blockquote>
| user8277998 | 516,330 | <p>Use parameterisation, </p>
<p>$$x^2+9y^2-4x+6y+4=(x-2)^2+(3y+1)^2- 1 = 0$$</p>
<p>Let $x - 2 = \cos t$ and $3y + 1 = \sin t$, then</p>
<p>$$f(t) := \dfrac{4x - 9y}{2} = \dfrac{4(\cos t + 2) - 3(\sin t - 1)}{2} = \dfrac{4\cos t - 3\sin t + 11}{2}$$</p>
<p>Equating first derivative to zero we get, $f^\prime(t) = ... |
1,972,002 | <p>Show an open set $A \subseteq \mathbb{R}$ contains no isolated points.</p>
<p>My attempt:
well if we can show an arbitrary element $a \in A$ is a limit point then we will be done. So since $A$ is open, there is an $\varepsilon > 0$ such that $B_\varepsilon(a) = (a - \varepsilon, a + \varepsilon) \subseteq A$. To... | Mark Joshi | 106,024 | <p>let $a_n = a + \epsilon/(2n)$ . This is a sequence in the set that converges to $a.$</p>
|
4,132,439 | <p>I know that <span class="math-container">$\int_1^\infty \frac1xdx$</span> diverges. I can probe that <span class="math-container">$\int_1^\infty \frac1{\ln(x)}dx$</span> diverges, as <span class="math-container">$\forall x>1:\frac1x<\frac1{\ln(x)}$</span></p>
<p>I also know that <span class="math-container">$\... | Yuta73 | 435,516 | <p>Nevermind, I made variable subtitution of <span class="math-container">$t=ln(x) \rightarrow e^tdt=dx$</span></p>
<p>So the integral ends up as
<span class="math-container">$$\int_0^\infty \frac{e^t}{t^7}dt$$</span>
But the limit of the integrand goes to infinity
<span class="math-container">$$\lim_{t\to+\infty} \fra... |
4,132,439 | <p>I know that <span class="math-container">$\int_1^\infty \frac1xdx$</span> diverges. I can probe that <span class="math-container">$\int_1^\infty \frac1{\ln(x)}dx$</span> diverges, as <span class="math-container">$\forall x>1:\frac1x<\frac1{\ln(x)}$</span></p>
<p>I also know that <span class="math-container">$\... | Sanjoy Kundu | 245,640 | <p>Take <span class="math-container">$x = e^u$</span>. Your integral transforms to <span class="math-container">$\int_0^\infty \frac{e^u}{u^7}du$</span>. You can now compare this with <span class="math-container">$\int_0^\infty \frac{1}{u^7}du$</span>, since <span class="math-container">$1 < e^u$</span> for all <sp... |
557,468 | <p>Let $f$ be a continuous map from $[0,1]$ to $[0,1].$ Show that there exists $x$ with $f(x)=x. $</p>
<p>I have $f$ being a continuous map from $[0,1]$ to $[0,1]$ thus $f: [0,1]\to [0,1]$. Then I know from the intermediate value theorem there exists an $x$ with $f(x)=x$ but I don't know how to formally prove it? </p>... | Community | -1 | <p>If it helps, you can think of this problem graphically as saying that for any function $f:[0,1]\to[0,1]$, $f$ must cross the diagonal of the square with vertices at $(0, 0),\ (1,1).$</p>
<p>Here's a picture:
<img src="https://i.stack.imgur.com/6ji5w.png" alt="enter image description here"></p>
<p>Hopefully it's cl... |
557,468 | <p>Let $f$ be a continuous map from $[0,1]$ to $[0,1].$ Show that there exists $x$ with $f(x)=x. $</p>
<p>I have $f$ being a continuous map from $[0,1]$ to $[0,1]$ thus $f: [0,1]\to [0,1]$. Then I know from the intermediate value theorem there exists an $x$ with $f(x)=x$ but I don't know how to formally prove it? </p>... | user104235 | 104,235 | <p>Can anyone check my answer to see if I understand this?</p>
<p>We have three cases: When $f(x) > x$ for every $x \in [0,1]$. When $f(x)< x$ for every $x \in [0,1]$ And finally, when $f(x)<x$ at some points of $x$ and $f(x)>x$ at other points of $x$. </p>
<p>If $f(x)> x$ at some points of $x\in [0,1]... |
2,555,811 | <p>Is there a name for the following type of ordering on some set $S$ {$a,b,c$} that includes only $>$ and $=$ for example: </p>
<p>$$a>b>c$$
$$a>b=c$$
$$a=b>c$$
$$a=b=c$$</p>
<p>Is there some name for these orderings? </p>
<hr>
<p>I know that all these <em>satisfy</em> a <em>total preorder</em> on $... | robjohn | 13,854 | <p>Since $e^t=1+t+O\!\left(t^2\right)$, set $t=\frac{\log(k)}x$
$$
\begin{align}
\lim_{x\to\infty}\left(\frac1n\sum_{k=1}^nk^{1/x}\right)^{nx}
&=\lim_{x\to\infty}\left(1+\frac1n\sum_{k=1}^n\frac{\log(k)}x+O\!\left(\frac1{x^2}\right)\right)^{nx}\\
&=\lim_{x\to\infty}\left(1+\frac1n\sum_{k=1}^n\frac{\log(k)}x\rig... |
2,372 | <p>How can I find the number of <span class="math-container">$k$</span>-permutations of <span class="math-container">$n$</span> objects, where there are <span class="math-container">$x$</span> types of objects, and <span class="math-container">$r_1, r_2, r_3, \cdots , r_x$</span> give the number of each type of object?... | Douglas S. Stones | 139 | <p>There's probably not going to be an easy way to do this... Consider two different examples of 15-letter "permutations". Then the number of permutations with that multiset of digits depends on the proportions of the chosen digits.</p>
<p>If you really want to do this, you can sum $k!/(s(1)! s(2)! \cdots s(x)!)$ (c... |
1,849,003 | <p>The following question was asked in an theoretical computer science entrance exam in India:</p>
<blockquote>
<p>A spider is at the bottom of a cliff, and is n inches from the top.
Every step it takes brings it one inch closer to the top with
probability $1/3$, and one inch away from the top with probability
... | Jaydeep Pawar | 761,596 | <p>The recurrence relation can be thought as,</p>
<p>T<sub>n</sub> = (1/3)<em>(T<sub>n-1</sub> + 1) + (2/3)</em>(T<sub>n-1</sub> + 1 + T<sub>n</sub> - T<sub>n-2</sub>)</p>
<p>On further simplification</p>
<p>T<sub>n</sub> = 3* T<sub>n-1</sub> - 2*T<sub>n-2</sub> + 3</p>
<p>The intuition is: To find expected number of s... |
602,248 | <p>This is the system of equations:
$$\sqrt { x } +y=7$$
$$\sqrt { y } +x=11$$</p>
<p>Its pretty visible that the solution is $(x,y)=(9,4)$</p>
<p>For this, I put $x={ p }^{ 2 }$ and $y={ q }^{ 2 }$.
Then I subtracted one equation from the another such that I got $4$ on RHS and factorized LHS to get two factors in te... | egreg | 62,967 | <p>You want
\begin{cases}
x=(7-y)^2\\
y=(11-x)^2\\
0\le x\le 11\\
0\le y\le 7
\end{cases}</p>
<p>The equation becomes
$$
x=49-14(121-22x+x^2)+(121-22x+x^2)^2
$$
which reduces to
$$
(x-9)(x^3-35x^2+397x-1444)=0
$$
(courtesy of WolframAlpha). The polynomial $f(x)=x^3-35x^2+397x-1444$ has at least one real root. It has ... |
4,461,482 | <p>In <span class="math-container">$ΔABC$</span>, if <span class="math-container">$(a+b+c)(a−b+c)=3ac$</span>, then which of the following is <span class="math-container">$\color{green}{\text{True}}$</span>?</p>
<ul>
<li><span class="math-container">$\angle B=60^\circ $</span></li>
<li><span class="math-container">$\an... | Wang YeFei | 955,668 | <p>You must show your attempts here first before any help can be given to you. As <strong>a hint</strong>: expand the left side and use the identity: <span class="math-container">$b^2=a^2+c^2-2ac\cos B$</span> to solve for <span class="math-container">$\cos B$</span> and then for <span class="math-container">$B$</span>... |
255,902 | <p>I have a double lattice sum and I was wondering how I could calculate this with Mathematica. In particular, I have a function <span class="math-container">$F:\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$</span> which takes as arguments a pair of points <span class="math-container">$x,y\in\mathbb{R}^2$</span>, and... | Domen | 75,628 | <p>You can use <code>Sum</code> to iterate twice through the set of lattice points.</p>
<pre><code>lattice = Catenate@Table[{x, y}, {x, 0, 3}, {y, 0, 3}]
(* {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, {1, 3}, {2,
0}, {2, 1}, {2, 2}, {2, 3}, {3, 0}, {3, 1}, {3, 2}, {3, 3}} *)
Sum[F[p1, p2], {p1, lattic... |
1,969,479 | <blockquote>
<p>Prove that the symmetric point of orthocenter $H$ of a triangle with respect to the midpoint of any side resides on the triangle's circumcircle.
<a href="https://i.stack.imgur.com/tkh4c.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/tkh4c.png" alt="enter image description here"><... | Community | -1 | <p>Use that BHCK is a quadrilateral whose diagonals intersect in the middle. So is a parallelogram. Then show that BHCK is inscribed.</p>
|
172,157 | <p>Today I was asked if you can determine the divergence of $$\int_0^\infty \frac{e^x}{x}dx$$ using the limit comparison test.</p>
<p>I've tried things like $e^x$, $\frac{1}{x}$, I even tried changing bounds by picking $x=\ln u$, then $dx=\frac{1}{u}du$. Then the integral, with bounds changed becomes $\int_1^\infty \f... | Robert Israel | 8,508 | <p>Comparison to $e^{x/2}$ should work (taking $x \to \infty$). So should $1/x$ (either as $x \to 0+$ or as $x \to \infty$). </p>
|
1,563,004 | <p>Assume we that we calculate the expected value of some measurements $x=\dfrac {x_1 + x_2 + x_3 + x_4} 4$. what if we dont include $x_3$ and $x_4$, but instead we use $x_2$ as $x_3$ and $x_4$. Then We get the following expression $v=\dfrac {x_1 + x_2 + x_2 + x_2} 4$.</p>
<p>How do I know if $v$ is a unbiased estimat... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. You may use
$$
(u \times v)'=u'v+uv'
$$ in the form
$$
\left(10^x\log_{10}(x)\right)'=\left(e^{x \ln 10} \times \frac{ \ln x}{\ln 10}\right)'.
$$ Can you take it from there?</p>
|
971,333 | <p>Given $\int \int dxdy$, I want to find the area bounded by $y=\ln\left(x\right)$ and $y=e+1-x$, and the $x$ axis. </p>
<p>I think the limits of integral in $y$ axis are from $y=\ln\left(x\right)$ to $y=e+1-x$ so $\int \limits_{y=\ln\left(x\right)}^{e+1-x}dy\int \:dx$, but I don't know how to find the limits of the ... | g.kov | 122,782 | <p>In this case it is convenient to use inverse representation of the curves,
$x=\mathrm{e}^y,\ x=\mathrm{e}+1-y$:</p>
<p><img src="https://i.stack.imgur.com/AFeBR.png" alt="enter image description here"></p>
<p>$$ \int_0^1 \int_{\mathrm{e}^y}^{\mathrm{e}+1-y} dx dy=\frac{3}{2}$$.</p>
|
17,285 | <p>I have problems in understanding few concepts of elementary set theory. I've choosen a couple of problems (from my problems set) which would help me understand this concepts. To be clear: it's not a homework, I'm just trying to understand elementary set's theory concepts by reading solutions.
<hr/>
<strong>Problem 1... | Jason DeVito | 331 | <p>I'll do 1 and 5.</p>
<p>For one, the first inclusion is true. For let $x\in f([a]_b)$. We want to show that $x\in [f(a)]_r$, i.e., that $x$ $r$ $f(a)$. Since $x\in f([a]_s)$, there is a $b\in A$ such that $b$ $s$ $a$ and f(b) = x. Since $b$ $s$ $a$, we know $f(b)$ $r$ $f(a)$. Thus, $x$ $r$ $f(a)$ so $x\in [f(a... |
4,642,772 | <p>I am trying to understand this proof:
<a href="https://math.stackexchange.com/questions/3201004/prove-that-varphipk-pk-pk-1-for-prime-p">Prove that $\varphi(p^k)=p^k-p^{k-1}$ for prime $p$</a></p>
<p>At some point it is stated that the number of multiples of p in range <span class="math-container">$[1,p^k)$</span> i... | Parcly Taxel | 357,390 | <p>Your powers-based counting only counts those numbers that are "pure" powers of <span class="math-container">$p$</span> – those numbers that when written in base <span class="math-container">$p$</span> have only one nonzero digit. Your counting omits numbers like <span class="math-container">$p+p^2$</span>.... |
975 | <p>The usual <code>Partition[]</code> function is a very handy little thing:</p>
<pre><code>Partition[Range[12], 4]
{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}
Partition[Range[13], 4, 3]
{{1, 2, 3, 4}, {4, 5, 6, 7}, {7, 8, 9, 10}, {10, 11, 12, 13}}
</code></pre>
<p>One application I'm working on required me to wri... | Mike Honeychurch | 77 | <p><strong>This is a complete re-write</strong></p>
<p>This is the original solution which was done in haste but i will leave here. It works in limited cases:</p>
<pre><code>multisegment[lst_List, scts_List, offset_List] :=
Module[{acc, offs},
offs = 1+Prepend[Accumulate[PadRight[offset,
1 + Ceiling[Lengt... |
975 | <p>The usual <code>Partition[]</code> function is a very handy little thing:</p>
<pre><code>Partition[Range[12], 4]
{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}
Partition[Range[13], 4, 3]
{{1, 2, 3, 4}, {4, 5, 6, 7}, {7, 8, 9, 10}, {10, 11, 12, 13}}
</code></pre>
<p>One application I'm working on required me to wri... | Mr.Wizard | 121 | <p>My first method was related to Mike's, yet unrefined. This is another method designed for speed. As written it only returns the portion of the list which can be partitioned into a complete set of partitions. This behavior can be changed with 4th+ argument of <code>Partition</code> and/or filtering.</p>
<pre><cod... |
3,361,224 | <blockquote>
<p>Show that a dense subspace <span class="math-container">$Y$</span> of a first countable separable topological space <span class="math-container">$X$</span> is separable.</p>
</blockquote>
<p><em>Proof:</em></p>
<p><span class="math-container">$X$</span> is separable.
Let <span class="math-container"... | Henno Brandsma | 4,280 | <p>This proof looks fine. Quite detailed. See Daniel's comment for an alternative faster proof.</p>
<p>To see that you need the first countable assumption on <span class="math-container">$X$</span>: if <span class="math-container">$X=[0,1]^\mathbb{R}$</span>, then <span class="math-container">$X$</span> is separable (... |
412,751 | <p>Let <span class="math-container">$G\subseteq \mathbb{C}^n$</span> be a bounded domain. Consider the Carathéodory metric <span class="math-container">$C_G$</span> on <span class="math-container">$G$</span>. If <span class="math-container">$G=\mathbb{D}^n$</span> (unit polydisc), then <span class="math-container">$C_G... | Jackson Morrow | 56,667 | <p>For a general formula, you should check out Kobayashi's book <em>Hyperbolic Complex Spaces</em>. In Proposition 3.1.11 of that text, he proves the following:</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be complex spaces. For <span class="math-container"... |
412,751 | <p>Let <span class="math-container">$G\subseteq \mathbb{C}^n$</span> be a bounded domain. Consider the Carathéodory metric <span class="math-container">$C_G$</span> on <span class="math-container">$G$</span>. If <span class="math-container">$G=\mathbb{D}^n$</span> (unit polydisc), then <span class="math-container">$C_G... | Peter Pflug | 343,739 | <p>In fact, there is the product property <span class="math-container">$\max\{c_X(x,x′),c_Y(y,y′)\}=c_{X×Y}((x,y),(x′,y′))$</span> for arbitrary domains <span class="math-container">$X, Y$</span>. See chapter 18 in "Invariant distances and metrics in complex analysis" de Gruyter (2013).</p>
|
50,479 | <p>Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?</p>
<p>x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).</p>
<p>If yes, will large number of solutions give moderate rank EC?</p>
<p>If one drops $... | Tapio Rajala | 11,716 | <p><em>Although this is not a mathematical answer I will put the results of my brute force search as an aswer as requested by jerr18. I didn't get anywhere with the thinking part.</em></p>
<h2>Code</h2>
<p>You can find the (non-optimal) C-code I wrote under <a href="http://users.jyu.fi/~tamaraja/temp/solu.c">my webpa... |
989,118 | <p>The sum is $$\sum_{n>0} \mathrm{i}^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$ and $$\sum_{n>0} (\mathrm{-i})^n\frac{J_n(x)}{n}\sin\frac{2\pi n}{3}\stackrel{?}{=}0$$</p>
<p>I suspect they are zero because I am working on a project which from symmetry consideration they should be zero. </p>
<p><st... | agha | 118,032 | <p>$\textbf{Hint:}$ Note that $i^{4n}-i^{4n+2}=2$ for $n \in \mathbb{N}$.</p>
|
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