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 |
|---|---|---|---|---|
1,520,643 | <p>If A² = I, prove that the matrix A is diagonalizable.</p>
<p>I have computed the eigenvalues to be 1 or -1 but I'm not sure how to proceed from here. </p>
<p>I'm thinking along the lines of "since rank(A + I) + rank(A - I) = n, therefore there exists n linearly independent vectors which corresponds to n eigenvecto... | Hirshy | 247,843 | <p>Hint: Let $\mu_A$ be the minimal polynomial of $A$. $A$ is diagonalizable iff $$\mu_A(x)=\prod\limits_{k=1}^m (x-x_k),m\leq n$$ with $x_i\neq x_j$ for $i\neq j$ being the eigenvalues of $A$.</p>
|
2,484,004 | <p><span class="math-container">$M^*(a,b)=b-a$</span> we know that this fact but how we can prove closed intervals are Lebesgue measurable. I tried to prove by using <span class="math-container">$\cap ((a-\frac1n),(b+\frac1n))$</span> But ı totaly stucked :( please help me guys</p>
| jonsno | 310,635 | <p>We have the function:
$$f(x,y) = \left(x+\frac{3}{4}\right)^2 + y^2 + \frac{7}{16}$$
This is a (imaginary?) <em>circle</em> with centre $C(-\frac{3}{4}, 0)$. The value of $f(x,y)$ at $P(x,y)$ depends on distance of $P$ from $C$ ($d^2 + 7/16$, where $d$ is distance of $P$ from $C$).</p>
<p>The other equation is an ... |
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | MCCCS | 357,924 | <p><span class="math-container">$$=\frac{\color{red}{\binom{9}{0}}+\color{blue}{\binom{9}{2}}+\color{orange}{\binom{9}{4}}+\color{green}{\binom{9}{6}}+\color{purple}{\binom{9}{8}}}{\color{red}{\binom{9}{0}}+\color{purple}{\binom{9}{1}}+\color{blue}{\binom{9}{2}}+\color{green}{\binom{9}{3}}+\color{orange}{\binom{9}{4}}+... |
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | ThisIsNotAnId | 24,567 | <p>Here's an analytical answer with greater emphasis on reasoning than anything specific to probability which might lend greater insight into the problem.</p>
<p>Consider if there was only one coin. The probability to have an even number of heads is <span class="math-container">$1\over2$</span>, since there are two po... |
2,406,061 | <p>I am also confused about whether these are symbols or have some meaning of their own.
PS- I know that <span class="math-container">$\operatorname{d}y\over\operatorname{d}x$</span> geometrically represents the slope. But, I've come across <span class="math-container">$\operatorname{d}x\over\operatorname{d}y$</span> t... | Ravenex | 442,239 | <p>In the terms dx and dy, the d is for delta or "change in". So they represent the change in y and the change in x as a function, usually in terms of each other but sometimes another parameter. So dy/dx as you said is the slope, or change in x divided by the change in y, dy/dx is simply the inverse slope.</p>
<p>The ... |
2,406,061 | <p>I am also confused about whether these are symbols or have some meaning of their own.
PS- I know that <span class="math-container">$\operatorname{d}y\over\operatorname{d}x$</span> geometrically represents the slope. But, I've come across <span class="math-container">$\operatorname{d}x\over\operatorname{d}y$</span> t... | P. Siehr | 457,090 | <p>Let's first start with $Δx$: </p>
<p>If you have two real numbers $x_0, x_1$, lets say $x_1>x_0$ you can calculate the different $x_1-x_0$. We define that difference as $Δx$:
$$Δx:=x_1-x_0$$
With that difference we can also write:
$$x_1 = x_0 + Δx.$$
or in words: If we add some change $Δx$ to $x_0$ we get $x_0+... |
1,291,107 | <p>Let $X$ be random variable and $f$ it's density. How can one calculate $E(X\vert X<a)$?</p>
<p>From definition we have:</p>
<p>$$E(X\vert X<a)=\frac{E\left(X \mathbb{1}_{\{X<a\}}\right)}{P(X<a)}$$</p>
<p>Is this equal to:</p>
<p>$$\frac{\int_{\{X<a\}}xf(x)dx}{P(X<a)}$$</p>
<p>? If yes, then ho... | Renato Faraone | 217,700 | <p>I know that the problem has already been answered but I want to show you a more general method, let's suppose that you don't se how to rewrite the equation:</p>
<p>$-x^4+16x^3-90x^2+199x-124=0$</p>
<p>Or I prefer to write:</p>
<p>$x^4-16x^3+90x^2-199x+124=0$</p>
<p>You can use something called the Ruffini rule: ... |
4,437,921 | <p>Given <span class="math-container">$X$</span>~<span class="math-container">$N(0,\sigma^2_X)$</span> and <span class="math-container">$Y$</span>~<span class="math-container">$N(0,\sigma^2_Y)$</span> (<span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are normally distributed random va... | RyRy the Fly Guy | 412,727 | <p>In response to the edit, note that <span class="math-container">$Z = \sin X \cos X = g(X)$</span> is a function <span class="math-container">$g$</span> of the normal random variable <span class="math-container">$X$</span>~<span class="math-container">$N(0,\sigma)$</span> whose pdf is given by <span class="math-conta... |
135,675 | <p>Let $D$ be the Dirac-Operator on $\mathbb{R}^n$ or more generally the Dirac spinor bundle $\mathcal{S}\to M$ of a (semi-)Riemannian spin manifold $M$. Then we consider $D$ as an unbouded Operator on $\mathcal{H}=L^2(\mathbb{R}^n)$ with domain $C^\infty_c(\mathbb{R}^n,\mathbb{C}^N)$. Then it is said that the operator... | paul garrett | 15,629 | <p>I think it is useful to ask the simpler question, why $f\cdot (1-\Delta)^{-1}$ is compact, on $\mathbb R^n$, when $f$ is a test function. Part of the point is that $\Delta$ itself (nevermind the Dirac operator) does <em>not</em> have compact resolvent on $\mathbb R^n$, essentially because Fourier inversion shows tha... |
70,803 | <p>Let $S$ be the sphere in $\mathbb{R}^3$ and $C:[0,1]\to S$ a continuously differentiable curve on $S$. Let $T:[0,1]\to\mathbb{R}^3$ denote the tangent vector of $C$. Let $P(t)$ be the plane containing $C(t)$ and having normal vector $T(t)$.</p>
<p>Given a size $d$ of the "paint brush" we define the "brush" $b:[0,1]... | Joseph O'Rourke | 6,094 | <p>The model is that used by Henryk Gerlach and Heiko von der Mosel in their 2010 paper "On sphere-filling ropes" <a href="http://arxiv.org/abs/1005.4609" rel="noreferrer">arXiv:1005.4609v1 (math.GT)</a> may be relevant.
Their question is different: What is the longest rope of a given thickness on a sphere?
But their e... |
991,878 | <p>How can it be proven that a cycle of length k is an even permutation if and only if k is odd?
I know it can be done using the fact that a permutation which exchanges two elements but leaves the rest unchanged is an odd permutation.</p>
| Paul | 17,980 | <p>Hints:</p>
<p>You should discuss that $\log a> 0$, $\log a<0$ and $\log a=0$ .</p>
|
991,878 | <p>How can it be proven that a cycle of length k is an even permutation if and only if k is odd?
I know it can be done using the fact that a permutation which exchanges two elements but leaves the rest unchanged is an odd permutation.</p>
| David K | 139,123 | <p>To follow up on Macavity's excellent answer, here's a proof that $x = 1$ is
the <em>only</em> value of $x$ that satisfies
$x \log a \leq a - 1$ for all positive real values of $a.$</p>
<p>Let $x \log a \leq a - 1.$
For $a > 1,$ we have $\log a > 0,$ and therefore
$$x \leq \frac{a - 1}{\log a}.$$
Now conside... |
2,002,201 | <p>simplify <span class="math-container">$\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}$</span>.</p>
<blockquote>
<p>1.<span class="math-container">$90^{\frac{3}{2}}$</span></p>
<p>2.<span class="math-container">$106\sqrt{41}$</span></p>
<p>3.<span class="math-container">$4\sqrt{41}$</span></p>
<p>4.<span class="ma... | fleablood | 280,126 | <p>Is this the nested radical formula?</p>
<p>$\sqrt{45 \pm 4\sqrt{41}} = a \pm b\sqrt{41}$</p>
<p>$45 \pm 4\sqrt{41} = (a^2 + 41b^2) \pm 2ab\sqrt{41}$</p>
<p>$a^2 + 41b^ = 45; 2ab = 4 \implies a=2;b = 1$</p>
<p>So $\sqrt{45 \pm 4\sqrt{41}} = |2 \pm \sqrt{41}|= \pm 2 + \sqrt{41}$</p>
<p>Plugging that into: $=(\sqr... |
4,510,384 | <p>In exercise 2.13 of page 43 of the book <a href="https://rads.stackoverflow.com/amzn/click/com/0134746759" rel="nofollow noreferrer" rel="nofollow noreferrer">Mathematical Proofs: A Transition to Advanced Mathematics</a> the reader is asked to state the logical negation of some statements. Of these, I find the autho... | Kuhlambo | 220,642 | <p>I hope it's possible to be logically consistent and easy to understand, I'll try to be both.</p>
<p>I think both versions of the sentence are equivalent, and I'll try to prove it, or at least to convince you.</p>
<p><strong>Hypothesis:</strong>
The sides are of different length.
<span class="math-container">$ \Leftr... |
4,510,384 | <p>In exercise 2.13 of page 43 of the book <a href="https://rads.stackoverflow.com/amzn/click/com/0134746759" rel="nofollow noreferrer" rel="nofollow noreferrer">Mathematical Proofs: A Transition to Advanced Mathematics</a> the reader is asked to state the logical negation of some statements. Of these, I find the autho... | ryang | 21,813 | <blockquote>
<p>"Two sides of the triangle have the same length."</p>
</blockquote>
<p>In other words, the triangle is isosceles (possibly equilateral).</p>
<p>So, the statement's <strong>negation</strong> is</p>
<ul>
<li><strong>Each side of the triangle has a distinct length.</strong></li>
<li><strong>Each ... |
3,357,697 | <p>I'm doing exercise II.4.5 in textbook Analysis I by Amann.</p>
<p>Could you please verify if my attempt contains logical mistakes/gaps! Thank you so much!</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/usrr7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/usrr7.png" alt="enter image de... | Badam Baplan | 164,860 | <p>McCoy's theorem extends as corollary to multivariate polynomials with very little work.</p>
<p>In <span class="math-container">$R[x, y] = R[x][y]$</span> a zero divisor <span class="math-container">$f$</span> is annihilated by an element <span class="math-container">$g\in R[x]$</span>, using McCoy's theorem.</p>
<p>... |
3,357,697 | <p>I'm doing exercise II.4.5 in textbook Analysis I by Amann.</p>
<p>Could you please verify if my attempt contains logical mistakes/gaps! Thank you so much!</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/usrr7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/usrr7.png" alt="enter image de... | Wuestenfux | 417,848 | <p>Well, use the lexicographic ordering on the polynomial ring <span class="math-container">$R=K[x_1,\ldots,x_n]$</span>.</p>
<p>Given two polynomials <span class="math-container">$f,g\ne 0$</span> in <span class="math-container">$R$</span>.
Suppose <span class="math-container">$x^\alpha$</span> and <span class="math-... |
451,722 | <p>I want to find the projection of the point $M(10,-12,12)$ on the plane $2x-3y+4z-17=0$. The normal of the plane is $N(2,-3,4)$.</p>
<p>Do I need to use Gram–Schmidt process? If yes, is this the right formula?</p>
<p>$$\frac{N\cdot M}{|N\cdot N|} \cdot N$$</p>
<p>What will the result be, vector or scalar?</p>
<p... | Tony Piccolo | 71,180 | <p>You can use calculus to minimize the distance (easier: the square of the distance) of M from the generic point of the plane.<br>
Use the equation of the plane to drop a variable, obtaining a function of two independent variables: compute the partial derivatives and find the stationary point. That's all.</p>
|
75,880 | <p>Say $f:X\rightarrow Y$ and $g:Y\rightarrow X$ are functions where $g\circ f:X\rightarrow X$ is the identity. Which of $f$ and $g$ is onto, and which is one-to-one?</p>
| Martin Sleziak | 8,297 | <p>You already have several answers which can help you remember the theorem. If you're looking for a proof (and have problems with showing it yourself), you might try to have a look at these links:</p>
<ul>
<li><a href="http://www.proofwiki.org/wiki/Injection_if_Composite_is_an_Injection" rel="nofollow noreferrer">htt... |
1,146,759 | <p>A covering of a group $G$ a family $\{S_i\}_{i \in I}$ of subsets of $G$ such that $G = \displaystyle \bigcup _{i \in I} S_i$.</p>
<p>Why is true that: A group covered by finitely many cyclic subgroups is either cyclic or finite?</p>
<p>Remark:
Is true that by Baer (see D. Robinson, Finiteness Conditions and Gener... | Hagen von Eitzen | 39,174 | <p>Some partial elementary results:</p>
<p>Assume $G$ is infinite and has a finite covering by finitely many cyclic subgroups $S_i=\langle s_i\rangle$, $i\in I$. We have to show that $G$ is cyclic. </p>
<p>If $S_i\subseteq S_j$ for some $i\ne j$, we may drop $S_i$. Hence we may assume wlog. that $S_i\not\subseteq S_j... |
1,598,545 | <p>Maybe I am not well versed with the actual definition of mean, but I have a doubt. On most resources, people say that arithmetic mean is the sum of $n$ observations divided by n. So my first question: </p>
<blockquote>
<p>How does this formula work? Is there any derivation to it? If not,
then while creating thi... | Kamil Jarosz | 183,840 | <p>You are talking about median, given some sequence of <strong>ordered</strong> numbers </p>
<p>$$1,1,2,2,3,3,\color{red}{3},6,6,7,8,9,9\tag{1}$$</p>
<p>The number at the center is called a median. When there are even number of elements,</p>
<p>$$1,1,2,2,3,\color{red}{3},\color{red}{6},6,7,8,9,9$$</p>
<p>Median is... |
3,845,475 | <p>Here's what I'm tasked with showing:</p>
<p>Let <span class="math-container">$(a_n)$</span> be a convergent sequence with <span class="math-container">$a_n\rightarrow a$</span> as <span class="math-container">$n\rightarrow\infty$</span>. By the Algebraic Limit Theorem, we know that <span class="math-container">$(a_n... | Dixon | 1,072,039 | <p>Given that the first ace is the 20th card to appear:
This suggests no Ace was chosen in the first 19 card flips. No other information was given, so every other card is subject to random choice in the first 19 card flips. On the 20th flip we are told an Ace is flipped. This gives the equation for P(A)</p>
<p><span ... |
1,477,058 | <p>Let $\textbf{$\gamma$}: \mathbb{R} \rightarrow \mathbb{R}^n$ be a smooth curve and let $T \in \mathbb{R}$. We say that $\textbf{$\gamma$}$ is $T$-periodic if $$\textbf{$\gamma$}(t+T)=\textbf{$\gamma$}(t) \text{ for all } t \in \mathbb{R}.$$
If $\textbf{$\gamma$}$ is not constant and is $T$-periodic for some $T\neq... | A.Γ. | 253,273 | <p>The period $2\pi$ is not right. Let's complexify $\gamma(t)=(x(t),y(t))$ as $x(t)+iy(t)$:
$$
\gamma=\cos^3(t)e^{i3t}=(\cos(t)e^{it})^3=\Bigl(\frac{e^{it}+e^{-it}}{2}e^{it}\Bigr)^3=\Bigl(\frac{z^2+1}{2}\Bigr)^3
$$
where $z=e^{it}$. The curve depends only on $z^2$ which is $\pi$ periodic, hence, the period is at most ... |
1,477,058 | <p>Let $\textbf{$\gamma$}: \mathbb{R} \rightarrow \mathbb{R}^n$ be a smooth curve and let $T \in \mathbb{R}$. We say that $\textbf{$\gamma$}$ is $T$-periodic if $$\textbf{$\gamma$}(t+T)=\textbf{$\gamma$}(t) \text{ for all } t \in \mathbb{R}.$$
If $\textbf{$\gamma$}$ is not constant and is $T$-periodic for some $T\neq... | Christian Blatter | 1,303 | <p>We had the same question a few days ago (<a href="https://math.stackexchange.com/questions/1478030/find-the-self-interection-differential-geometry/1478134#1478134">Find the self-interection.Differential Geometry</a>). I'm extending my accepted answer there to a full solution.</p>
<p>Write your curve in the form
$$\... |
1,477,058 | <p>Let $\textbf{$\gamma$}: \mathbb{R} \rightarrow \mathbb{R}^n$ be a smooth curve and let $T \in \mathbb{R}$. We say that $\textbf{$\gamma$}$ is $T$-periodic if $$\textbf{$\gamma$}(t+T)=\textbf{$\gamma$}(t) \text{ for all } t \in \mathbb{R}.$$
If $\textbf{$\gamma$}$ is not constant and is $T$-periodic for some $T\neq... | amcalde | 168,694 | <p>Just using trig:</p>
<p>$T = \pi$</p>
<p>see this by computing
$\gamma(0) = (1,0) = \gamma(T)$</p>
<p>Solve this using the second component first so that $$0 = \textrm{sin}(3T)\textrm{cos}^3(T)$$
Then $T = n \pi/3$, Do the same thing for the first component and you must have $n$ is a multiple of 3. In fact this ... |
320,557 | <p>let $S$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show this is a topological manifold. for starters one needs to define a suitable topology on it. i was thinking let a set $U$ be open in $S$ iff $U \cap S^2$ (intersection with sphere) is open in $S^2$ in the s... | Albert | 19,331 | <p>It is a smooth manifold of dimension 3.
Indeed : let $F : S^2 \times S^2 \rightarrow \mathbb{R}$ be defined by $F(x,y)=\sum_i x_i y_i$ (the standard inner product). $F$ is a smooth map on the smooth manifold $S^2 \times S^2$ and its derivative does not vanish so it is a submersion. Bu the submersion property, the s... |
18,960 | <p>I am making this post in regards to the ongoing delete/undelete skirmish (let's at least change the monotonicity of the use of "war"). The old version of the question is <a href="https://math.stackexchange.com/revisions/172652/3">here</a>, the current version (after edits today) <a href="https://math.stackexchange.c... | Thomas | 26,188 | <p>I might take a controversial stand on this one. This is just my opinion.</p>
<p>The original question was a statement only question and should have been closed and deleted despite the existence of a good answer. As a principle I think it is wrong for anyone (even a moderator) to radically change the question withou... |
4,580,470 | <p>Suppose <span class="math-container">$X$</span> is a Geometric random variable (with parameter <span class="math-container">$p$</span> and range <span class="math-container">$\{k\geq 1\}$</span>).</p>
<p>Let <span class="math-container">$M$</span> be a positive integer.</p>
<p>Let <span class="math-container">$Z:=\m... | Siong Thye Goh | 306,553 | <p>The equation holds for nonnegative random variable.</p>
<p><span class="math-container">\begin{align}
E[Z]&=\sum_{k=1}^\infty kP[Z=k]\\
&=P(Z=1)\\
&+P(Z=2) + P(Z=2)\\
&+P(Z=3) + P(Z=3) + P(Z=3)+\ldots\\
\vdots\\
&=P(Z\ge 1) + P(Z\ge 2)+P(Z\ge 3)+\ldots\\
&=\sum_{k=1}^\infty P(Z \ge k)
\end{al... |
2,547,933 | <p>Consider the integral $I=\displaystyle\int_{R}\int f(x,y)dx dy$ over the region $R$, given by the triangle with vertices $(0,0),(1,1)$ and $(2,0)$. </p>
<p>This is an isosceles triangle with one side lying along the $x-$axis. So, our domain is not "nice" to find the bounds for integral I assume, since even if we w... | Mark Bennet | 2,906 | <p>Expand both using the binomial theorem</p>
<p>$$\left(1+\frac 1n\right)^n=1^n+n\cdot\frac 1n\cdot1^{n-1}+\binom n2\left(\frac 1n\right)^21^{n-2}+\binom n3\left(\frac 1n\right)^31^{n-3}+\dots =$$$$=1+1+\frac {n(n-1)}{2n^2}+\frac {n(n-1)(n-2)}{6n^3}+\dots$$while $$\left(1-\frac 1n\right)^n=1-1+\frac {n(n-1)}{2n^2}-\f... |
920,782 | <p>How do I find the number of integral solutions to the equation - </p>
<p>$$2x_1 + 2x_2 + \cdots + 2x_6 + x_7 = N$$</p>
<p>$$x_1,x_2,\ldots,x_7 \ge 1$$</p>
<p>I just thought that I should reduce this a bit more, so I replace $x_i$ with $(y_i+1)$, so we have:</p>
<p>$$y_1 + y_2 + \cdots + y_6 = \tfrac{1}{2}(N + 13... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
504,524 | <p>I'm trying to learn probability and statistics but I can't really get my head around this one. I realize after drawing the first card there will only be 51 cards in the deck but I'm having trouble calculating the chance that the second one is an Ace if I don't know what the first card is?</p>
<p>Assuming that the i... | Community | -1 | <p>Hint: If the first one you draw is an Ace then the probability for the second one to be an Ace is $\frac3{51}$;</p>
<p>however if the first one drawn is <strong>not</strong> an Ace then the probability is $\frac4{51}$.</p>
|
504,524 | <p>I'm trying to learn probability and statistics but I can't really get my head around this one. I realize after drawing the first card there will only be 51 cards in the deck but I'm having trouble calculating the chance that the second one is an Ace if I don't know what the first card is?</p>
<p>Assuming that the i... | kiss my armpit | 26,975 | <p>There are $52\times 51$ outcomes when you draw 2 cards one after another without returning the first drawn card to the deck.</p>
<p>There are $48\times 4$ outcomes when the first card is not Ace and the second card is Ace.
There are $4\times 3$ outcomes when both cards are Ace. The total outcomes are $48\times 4 + ... |
3,762,624 | <p>This is a pretty common question in probability and there are already a few answers on the site. HOWEVER, my Professor has put a little twist on it and I can't piece it together anymore.</p>
<p><strong>QUESTION:</strong></p>
<p>Let <span class="math-container">$(X_n)$</span> be a sequence of geometric random variabl... | Ben Grossmann | 81,360 | <p><strong>Counterexample:</strong> Take <span class="math-container">$t = 1$</span> and define
<span class="math-container">$$
g(x) = x^2, \quad f(x) = 2x^2 - \frac 13 x^3.
$$</span>
In particular, the ratios come out to
<span class="math-container">$$
\frac{tg(t)}{\int_0^tg(s)\,ds} = 3, \quad
\frac{tf(t)}{\int_0^tf(... |
337,930 | <p>Given two polynomials</p>
<p>$$
p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \\
q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n}
$$</p>
<p>And the series expansion from their rational polynomial</p>
<p>$$
\frac{p(x)}{q(x)} = c_0 + c_1 x + c_2 x^2 + \ldots
$$</p>
<p>is it possible to recover the the o... | Rohan Shinde | 463,895 | <p>It can be proven combinatorially by noting that any combination of $r$ objects from a group of $m+n$ objects must have some $0\le k\le r$ objects from group $m$ and the remaining from group $n$.</p>
|
622,076 | <p>Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? </p>
<p>It seems to me like they are equal definitions in a way. </p>
<p>Can you give me a counter-example? </p>
<p>Thanks</p>
| DanZimm | 37,872 | <p>To begin I want to state the IVP considering I messed up on the definition:</p>
<blockquote>
<p>Let $I$ be an open interval and $f : I \to \mathbb{R}$ then $f$ has the IVP iff Given $a,b \in I : a \le b$
$$
\forall \; y \text{ between } f(a),f(b) \; \exists \; x \in [a,b] : f(x) = y
$$</p>
</blockquote>
<p>The... |
622,076 | <p>Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? </p>
<p>It seems to me like they are equal definitions in a way. </p>
<p>Can you give me a counter-example? </p>
<p>Thanks</p>
| N. S. | 9,176 | <p>The next theorem might be of interest to you, it really shows that the class of functions with the IVP is very big.</p>
<p><strong>Theorem (Sierpinski)</strong> Let $f : \mathbb R \to \mathbb R$ be any function. Then there exists $f_1,f_2 : \mathbb R \to \mathbb R$ such that $f=f_1+f_2$ and $f_1,f_2$ satisfy the In... |
442,950 | <p>I would like to show <span class="math-container">$\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$</span>.</p>
<p>Now, of course, the integrand does not converge uniformly to <span class="math-container">$0$</span> on <span class="math-container">$\theta\in [0, \pi/2]$</span>, since it has... | Potato | 18,240 | <p>Split it into two pieces: one integral over $[0,\epsilon]$ and another over $[\epsilon, \pi/2]$. Since the integrand is bounded on the first piece, you can make it arbitrarily small by choosing $\epsilon$ small. On the other piece, it converges uniformly to zero. I think you can take it from there. </p>
|
2,836,539 | <p>Recently I run into this integral</p>
<p>$$\mathcal{J} = \int_{0}^{\pi/2} \frac{x \log \left ( 1-\sin x \right )}{\sin x} \, \mathrm{d}x$$</p>
<p>I don't know to what it evaluates. I tried several approaches.</p>
<p><strong>1st:</strong> Differentiation under the integral sign</p>
<p>Consider the function $\disp... | Quanto | 686,284 | <p>After the substitution <span class="math-container">$t=\tan\frac x2$</span>, it can be shown that the integral reduces to</p>
<p><span class="math-container">\begin{align}
& \int_{0}^{\pi/2} \frac{x \log \left ( 1-\sin x \right )}{\sin x} \, {d}x\\
=&\> 4\int_0^1 \frac{\tan^{-1}t\ln t}tdt
= -2\int_0^1 \f... |
3,832,383 | <p>Below is a problem I found, however, after many attemps I can not seem to get a solution.</p>
<p><strong>Problem:</strong> Let <span class="math-container">$E \subset [0,1]$</span>. Show that if <span class="math-container">$m^*(E) + m^*([0,1] \setminus E) = 1$</span>, then <span class="math-container">$E$</span> is... | Angelo | 771,461 | <p><span class="math-container">$(2)\implies(1)$</span></p>
<p><strong>Proof:</strong></p>
<p>Assume that <span class="math-container">$\;E\;$</span> is a subset of <span class="math-container">$\;[0,1]\;$</span> such that (2) holds.</p>
<p>Let <span class="math-container">$\;Y\;$</span> be a <span class="math-containe... |
871,542 | <p>I have the following theorem:</p>
<blockquote>
<p>Let <span class="math-container">$\rho$</span> be the traffic intensity.</p>
<p>a) If <span class="math-container">$\rho<1$</span>, then <span class="math-container">$X$</span> is positive recurrent.</p>
<p>b) If <span class="math-container">$\rho>1$</span>, t... | Mark Fantini | 88,052 | <p>He mentions that $Z(t) = e^{it}.$ You have $$|Z(t)| = |e^{it}| = \sqrt{\cos^2(t) + \sin^2(t)} = 1.$$</p>
|
2,780,731 | <p>In school, I have recently been learning about simple differential equations. We know that the solution of $y'=y$ is $y=Ae^x$, where $A$ is a constant. But how can we know that it is the <strong>only</strong> solution? The only thing I can figure out is that $y$ is continuously differentiable. Help me, please.</p>
| Mohammad Riazi-Kermani | 514,496 | <p>Suppose $y$ is a solution to $y'=y$</p>
<p>Multiply both sides by $e^{-x}$ to get $$ y'e^{-x} = ye^{-x}$$</p>
<p>$$y'e^{-x}-ye^{-x}=0$$</p>
<p>$$ \frac {d}{dx} (ye^{-x}) =0$$
$$ye^{-x} =A$$
$$ y=Ae^{x} $$</p>
|
1,598,006 | <p>(Here, $B$ is relatively compact means the closure of $B$ is compact.)</p>
<blockquote>
<ol>
<li><p>$\hat A$ is compact.</p></li>
<li><p>$\hat A=\hat {\hat A}$.</p></li>
<li><p>$\hat A$ is connected.</p></li>
<li><p>$\hat A=X$.</p></li>
</ol>
</blockquote>
<p>I try to eliminate the options by using an ... | DanielWainfleet | 254,665 | <p>For (1) let $X=R$ and $A=(0,1)$. The connected components of $X-A$ are $(-\infty,0]$ and $[1,\infty)$ which are closed in $X$ but not compact.So $\hat A=A$, which is not compact.For (3) let $X=R^2$ and let $A$ contain exactly 2 points.Then $X-A$ is connected but $\overline {X-A}=X$ is not compact. So $\hat A=A$, whi... |
1,600,063 | <p>I am trying to prove the following statement:</p>
<p>Given any two real numbers $x,y$ with $x<y$, there exists a rational number $q$ that satisfies $x<q<y$.</p>
<p>I got stuck at one point of the proof, so this is what I thought of:</p>
<p>I want to find a rational number $q$, which can be expressed as $... | Hagen von Eitzen | 39,174 | <p>Let $a$ be the least integer $>bx$ and show that then $a<by$, for otherwise $a-1$ would also be $>bx$.</p>
|
3,752,771 | <p>I wanted to get the full probability of 2 attempts made at 60% chance of success.</p>
<p>I was looking at a different chain of math and found my probability to hit an enemy is 60% per each attack but I was wondering how it would look at all the outcomes and the probability of it.</p>
<blockquote>
<p>6/10 * 6/10 = 36... | Alex | 38,873 | <p>You got 136% by adding <span class="math-container">$60\%+40\%+36\%$</span>, i.e. <span class="math-container">$P(S) + P(F) + P(SS)$</span>. This is incorrect because these outcomes do not partition sample space (full set of outcomes).</p>
<p>Your sample space are the following outcomes:
<span class="math-container"... |
2,481,046 | <p>I have a question that asks to show that $S^2 = \{(x,y,z) \in \mathbb{R}^3|x^2+y^2+z^2=1\}$ is a differentiable manifold. My professor says that one way to do this is to define the following 6 parametrizations of the sphere, which cover the entire sphere.</p>
<p>$\vec{\phi_{i}}:V \to \mathbb{R}^3$ where $V = \{(u,v... | Jack D'Aurizio | 44,121 | <p>The function $\sin\left(x+\frac{1}{x}\right)$ is continuous and bounded between $-1$ and $1$ on the interval $(0,1)$, hence the given integral is trivially convergent</p>
<p>By enforcing the substitution $x+\frac{1}{x}\mapsto z$ we have</p>
<p>$$\begin{eqnarray*} \int_{0}^{1}\sin\left(x+\tfrac{1}{x}\right)\,dx &am... |
2,456,976 | <p>$ f(x,y) = \begin{cases} \dfrac{x^3+y^3}{x^2+y^2} &\quad\text{if} [x,y] \neq [0,0]\\[2ex] 0 &\quad\text{if}[x,y] = [0,0]\\ \end{cases} $</p>
<p>The only point it could be discontinuous in is <code>[0,0]</code>. How do I find the limit of the function for $(x,y) \rightarrow (0,0)$? $ \lim_{(x,y) \rightarrow... | Yaddle | 333,729 | <p>For $x,y \in \mathbb R \setminus \{0\}$ we have</p>
<p>$$\vert f(x,y)\vert = \left\vert\frac{x^3 + y^3}{x^2+ y^2}\right\vert \leq \left\vert \frac{x^3}{x^2+ y^2} \right\vert + \left\vert \frac{y^3}{x^2+ y^2}\right\vert \leq \left\vert \frac{x^3}{x^2} \right\vert + \left\vert \frac{y^3}{y^2}\right\vert = \vert x \ve... |
1,105,454 | <p>We have $f: \Bbb{R} \rightarrow \Bbb{R}$ defined as follows:</p>
<p>$$f(x) = \begin{cases} a, & \mbox{if } x=0 \\ \sin\frac{b}{|x|}, & \mbox{if } x\neq 0 \end{cases}$$</p>
<p>The problem asks us to tell for which $a,b \in \Bbb{R}$, $f$ is continuous.</p>
<p>Intuitively I should find $\lim_{x\rightarrow 0}... | hmakholm left over Monica | 14,366 | <p>Speaking of the "value" or "result" of the operation would by far be the most understandable.</p>
|
1,105,454 | <p>We have $f: \Bbb{R} \rightarrow \Bbb{R}$ defined as follows:</p>
<p>$$f(x) = \begin{cases} a, & \mbox{if } x=0 \\ \sin\frac{b}{|x|}, & \mbox{if } x\neq 0 \end{cases}$$</p>
<p>The problem asks us to tell for which $a,b \in \Bbb{R}$, $f$ is continuous.</p>
<p>Intuitively I should find $\lim_{x\rightarrow 0}... | MphLee | 67,861 | <p>Generalizing the question to functions, note that $\circ$ is just a <em>binary function</em> $\circ(x,y)=z$ but expressed with <a href="http://en.wikipedia.org/wiki/Infix_notation" rel="nofollow"><em>infixed notation</em></a> $x\circ y$, in my opinion the common terminology is: <em>given a</em> $n$*-ary function* $f... |
2,661,468 | <p>the number of not identically zero functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation $f(xy)=f(x)f(y)$ and $f(x+z)=f(x)+f(z)$ for some $z$ not equal to zero</p>
<ol>
<li>one</li>
<li>finite</li>
<li>countable</li>
<li>uncountable</li>
</ol>
<p>it seems like the question asks about the number of homomo... | lhf | 589 | <p>Since $f$ is not identically zero, take $y$ such that $f(y)\ne0$. Then $f(y)=f(1y)=f(1)f(y)$ implies $f(1)=1$.</p>
<p>Take $x=0$ in $f(x+z)=f(x)+f(z)$ and get $f(z)=f(0)+f(z)$, which implies $f(0)=0$.</p>
<p>Therefore, $f$ is a ring homomorphism $\mathbb{R} \to \mathbb{R}$.</p>
<p>Now see <a href="https://math.st... |
891,575 | <p>The circumference of a circle has length 90 centimeters, Three points on the circle divide the circle into three equal lengths. Three ants A, B, and C start to crawl clockwise on the circle, with starting from one of the three points. Initially A is ahead of B and B is ahead of C. Ant A crawls 3 centimeters per seco... | DJohnM | 58,220 | <p>Another way of simplifying the problem:</p>
<p>A is at 4 o'clock, B is at 12 o'clock, and C is at 8 o'clock.</p>
<p>Let's subtract $5$ cm/s from each ant's speed. So now C is going clockwise at only $5$ cm/s, B is stationary, and A is going <strong>counter-clockwise</strong> at $2$ cm/s. And they're all still $3... |
630,966 | <p>Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most of these metric spaces are normed vector spaces. Why not just cover normed vector spaces instead of metric spaces? </... | Ittay Weiss | 30,953 | <p>Metric spaces are far more general than normed spaces. The metric structure in a normed space is very special and possesses many properties that general metric spaces do not necessarily have. </p>
<p>Metric spaces are also a kind of a bridge between real analysis and general topology. With every metric space there ... |
2,694,740 | <p>$$\frac{2.10^{-7} - 0,4.10^{-6}}{10^{-8}} = ? $$</p>
<p>These questions are making me confused because we're dealing with the terms like $10^x$. What are your professional tips? </p>
<p><strong>My attempt:</strong></p>
<p>$$\frac{2.10^{-7} - 4.10^{-7}}{10^{-8}} \tag{1} $$
$$\frac{ -8.10^{-7}}{10^{-8}} \tag{2} $$<... | Dr. Sonnhard Graubner | 175,066 | <p>multplying numerator and denominator by $10^{8}$ we get
$$2\cdot 10^{-7}\cdot 10^{8}-0.4\cdot 10^{-6}\cdot 10^{8}$$ and we get $$2\cdot 10-0.4\cdot 10^2=...$$</p>
|
2,867,404 | <p>I have a problem how to get the area from the picture.
Some ideas I got are not good enough to get the correct value of the whole element.</p>
<p><a href="https://i.stack.imgur.com/PeQ5O.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PeQ5O.jpg" alt="enter image description here"></a></p>
| Mladen Uzelac | 580,641 | <p>I asked someone and this answer will satisfy me</p>
<p><a href="https://i.stack.imgur.com/Qsq8g.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Qsq8g.jpg" alt="enter image description here"></a></p>
|
87,466 | <p>I have some code which involves tiny numbers being put to the power of very large numbers. The function I'm looking at is</p>
<p>$\varphi = \omega(T) \left(1 - (1 - \epsilon)^{n_{e}(T)} \right)$</p>
<p>when $\epsilon $ is very small (~$10^{-16}$) and $n_{e}$ is large ($> 10^{10}$). Both $n_{e}$ and $\omega$ are... | Fred Simons | 20,253 | <p>As J. M. already suggested, this is a machine precision issue. So let us do your computation with arbitrary precision numbers. For doing so, all machine numbers have to be replaced with arbitrary precision numbers, otherwise the computation falls back to machine numbers. In the following command I have done this by... |
3,995,728 | <p>For the integral
<span class="math-container">$$
\int_1^2 \frac {3^x + 2}{3^{2x} + 3^x} dx
$$</span>
choose the interval of its result: <span class="math-container">$(-\infty, 0], (0, \frac 12], (\frac 12, 1], (1, 3], (3, \infty)$</span>. According to the author of this task you do not have to compute the integral i... | Ethan Bolker | 72,858 | <p>Hint. In the upper circle you have the information you need to find the <span class="math-container">$\sin$</span> of half the central angle.</p>
|
441,792 | <p>There are many objects in mathematics that have the term "chiral" in their name, for instance, chiral algebra by Beilinson and Drinfeld, chiral de Rham complex, chiral Koszul duality etc. Some people told me that chiral algebras are <span class="math-container">$2$</span>-dimensional analogue of associativ... | Carlo Beenakker | 11,260 | <p><strong>Q:</strong> <em>What is the geometry/physics behind the term chiral?</em></p>
<p>In the physics context, a Hamiltonian <span class="math-container">$H$</span> is said to possess chiral symmetry if it anticommutes with a unitary involution <span class="math-container">$C$</span>. Eigenstates <span class="math... |
1,419,315 | <p>I have a particular scenario.</p>
<p>In this scenario, we have the standard cubic equation,</p>
<pre><code>ax^3 + bx^2 + cx + d = y
</code></pre>
<p>as well as 3 points that are graphed, <a href="https://i.imgur.com/VCZKuGW.png" rel="nofollow noreferrer">as can be seen in this graph</a>. (The line is irrelevant ... | Eugene Zhang | 215,082 | <p>A simple way is to calculate all principle minors of <span class="math-container">$A$</span>. If they are all positive, then <span class="math-container">$A$</span> is positive definite.</p>
<p>For example, <span class="math-container">$|A|_1=2>0$</span></p>
<p><span class="math-container">$$
|A|_2=\left|\begin{a... |
509,635 | <p>If every chain in a lattice is complete (we take the empty set to be a chain), does that mean that the lattice is complete? If yes, why? </p>
<p>My intuition says yes, and the reasoning is that we should somehow be able to define a supremum of any subset of the lattice to be the same as the supremum of some chain r... | Doug | 725,720 | <p>By completeness, I mean every nonempty subset has a sup and inf. I am not sure if a chain-complete poset is complete and my personal guess is negative. Here is a proof showing a nonempty chain-complete lattice <span class="math-container">$L$</span> is complete, with the lattice property emphasised.</p>
<p>By Zorn's... |
3,537,654 | <p><span class="math-container">$$\lim_{x\to 0^{+}} (\tan x)^x$$</span></p>
<p><span class="math-container">$$\lim_{x\to 0^{+}} e^{\ln((\tan x)^x)}=\lim_{x\to 0^{+}} e^{x\ln(\tan x)}=\lim_{x\to 0^{+}} e^{x[\ln(\sin x)-\ln(\cos x)]}$$</span></p>
<p>We can continue to create an expression that may help us use L'Hospita... | Bernard | 202,857 | <p>Compute first the limit of the logarithm, using <em>equivalence</em>:</p>
<p><span class="math-container">$\tan x\sim_0 x,\:$</span> so
<span class="math-container">$\quad \ln\bigl((\tan x)^x\bigr)=x\ln(\tan x)\sim_0x\ln x \xrightarrow[x\to 0]{} 0.$</span></p>
|
3,537,654 | <p><span class="math-container">$$\lim_{x\to 0^{+}} (\tan x)^x$$</span></p>
<p><span class="math-container">$$\lim_{x\to 0^{+}} e^{\ln((\tan x)^x)}=\lim_{x\to 0^{+}} e^{x\ln(\tan x)}=\lim_{x\to 0^{+}} e^{x[\ln(\sin x)-\ln(\cos x)]}$$</span></p>
<p>We can continue to create an expression that may help us use L'Hospita... | trancelocation | 467,003 | <p>You surely know</p>
<ul>
<li><span class="math-container">$\lim_{x\to 0}\frac{\sin x}{x} = 1 \Rightarrow \lim_{x\to 0}\frac{\tan x}{x} = \lim_{x\to 0}\left(\frac{\sin x}{x}\cdot \frac 1{\cos x} \right) = 1$</span>.</li>
<li>Besides this, it is easy to show that <span class="math-container">$\lim_{x\to 0^+}x^x = 1$<... |
3,371,964 | <p>Let be <span class="math-container">$O_{2}$</span> the orthogonal group, that is, the group of reflections and rotations of <span class="math-container">$\mathbb{R}^{2}$</span>. His center is <span class="math-container">$\{ \pm I\} \simeq \mathbb{Z}_{2}$</span>. I'm having problems to study the center of the quotie... | Inácio | 400,062 | <p>I was ignoring the part "anticommutes with every orthogonal matrix". Indeed, we can look to commutators:</p>
<p><span class="math-container">\begin{align}
[Rot(\phi), Ref(\psi)] &= Rot(-2\phi)\\
[Rot(\phi), Rot(\psi)] &= I_{2}\\
[Ref(\phi), Ref(\psi)] &= Rot(4(\phi - \psi))
\end{align}... |
1,263,865 | <p>So I have that $700=7\cdot2^2\cdot5^2$ and I got that $3^2\equiv1\pmod2$ so then $3^{1442}\equiv1\pmod2$ also $3^2\equiv1\pmod{2^2}$ so $3^{1442}\equiv1\pmod{2^2}$ which covers one of the divisors of $700$. Im not sure if I'm supposed to use $2$ or $2^2$ and I was able to find that $3^2\equiv-1\pmod5$ so $3^{1442}\e... | Adhvaitha | 228,265 | <p>Go $\pmod4$, $\pmod7$ and $\pmod{25}$. We have
\begin{align}
3^2 \equiv 1\pmod4\\
3^6 \equiv 1\pmod7\\
3^{20} \equiv 1\pmod{25}
\end{align}
This gives us that
\begin{align}
3^{60} \equiv 1\pmod4\\
3^{60} \equiv 1\pmod7\\
3^{60} \equiv 1\pmod{25}
\end{align}
This means
$$3^{60} \equiv 1\pmod{700}$$
Note that $3^{1442... |
225,128 | <p>I'm trying to define a function accepting only real values like this:</p>
<pre><code>f[x_Real] := x^2
f[0]
</code></pre>
<p>But it outputs</p>
<blockquote>
<p><code>f[0] </code></p>
</blockquote>
<p>and doesn't output 0.</p>
<p>Is there any reason why <code>f[x_Real]</code> doesn't work? I tested <code>f[x_Integer]<... | Natas | 67,431 | <p>Let me explain what went wrong.</p>
<p>When you define a function <code>f[pattern] := ...</code> then this will check at every occurrence of <code>f[x]</code> if <code>x</code> matches <code>pattern</code>.</p>
<p>Your pattern was <code>x_Real</code> which means "match any expression whose <code>Head</code> is ... |
2,204,944 | <p>A line is a collection of infinitely many points. By definition, a point has no dimensions. But, how can infinitely many dimensionless points give rise to a line with a dimension. This is the same case with planes, solids and higher dimensions too...</p>
<p>Thanks in advance for any help..!!</p>
| Graham Kemp | 135,106 | <p><strong>Long story short:</strong> The random variables have identical expectations when they follow <em>identical distributions</em>.</p>
<hr>
<p>I have a deck of these four cards, with values 1,2,3,4. I shuffle the deck and place two cards face down, one on your left and one on your right. </p>
<p>What i... |
834,228 | <p>$$u_{1}=2, \quad u_{n+1}=\frac{1}{3-u_n}$$
Prove it is decreasing and convergent and calculate its limit.
Is it possible to define $u_{n}$ in terms of $n$?</p>
<p>In order to prove it is decreasing, I calculated some terms but I would like to know how to do it in a more "elaborated" way.</p>
| Claude Leibovici | 82,404 | <p>If there is a limit, it will be defined by $$L=\frac{1}{3-L}$$ which reduces to $L^2-3L+1=0$. You need to solve this quadratic and discard any root greater than $2$ since this is the starting value and that you proved that the terms are decreasing.</p>
|
834,228 | <p>$$u_{1}=2, \quad u_{n+1}=\frac{1}{3-u_n}$$
Prove it is decreasing and convergent and calculate its limit.
Is it possible to define $u_{n}$ in terms of $n$?</p>
<p>In order to prove it is decreasing, I calculated some terms but I would like to know how to do it in a more "elaborated" way.</p>
| JimmyK4542 | 155,509 | <p>As others have said, you don't need a formula for $u_n$ to show that it is strictly decreasing and converges to $\dfrac{3-\sqrt{5}}{2}$. </p>
<p>However, You can get an exact formula for $u_n$. Lets assume $u_n = \dfrac{a_n}{b_n}$ for integers $a_n, b_n$. </p>
<p>Then, $\dfrac{a_{n+1}}{b_{n+1}} = u_{n+1} = \dfrac{... |
3,964,429 | <p>Zeckendorf : <em>Every positive integer N can be expressed uniquely as a sum of distinct non-consecutive Fibonacci numbers</em></p>
<p>I was wondering if this theorem can be applied with the extended Fibonacci numbers, and especially I am looking for a way to <strong>find the Zeckendorf-like representation of <span ... | Crypto | 234,832 | <p><a href="https://www.fq.math.ca/Scanned/30-2/bunder.pdf" rel="nofollow noreferrer">This paper</a> that proves the existence and shows a method</p>
<p><em>(we prove that) every integer can be represented uniquely as a sum of nonconsecutive Fibonacci numbers Fi where i < 0 and we specify an algorithm that leads to ... |
4,206,039 | <p>Find the radius of convergence of the following power series <span class="math-container">$$\sum_{n=1}^\infty \frac{(-1)^n z^{n(n+1)}}{n}$$</span></p>
<p>Here's my working
<span class="math-container">$$\lim_{n\to \infty}| \frac{(-1)^{n+1} z^{(n+1)(n+2)}}{n+1} \frac{n}{(-1)^nz^{n(n+1)}}|$$</span>
<span class="math... | DonAntonio | 31,254 | <p>You could try to do the following to see your series as a "usual" power series:</p>
<p><span class="math-container">$$\frac{(-1)^n z^{n(n+1)}}n=a_mz^m\;,\;\;\text{when}\;\;a_m=\begin{cases}0,&m\neq2,6,12,...,k(k+1)\\{}\\\cfrac{(-1)^m}m,&m=k(k+1)\end{cases}\;,\;\;k\in\Bbb N$$</span></p>
<p>From here... |
2,073,923 | <p>It seems the number of nonnegative integer solutions to the equation $xyz=n$ is given by
$$\sum\limits_{d \mid n} \tau(d)$$</p>
<p>$\tau$ is the number of divisors function. I'm wondering if there is a way to simplify this sum. Really appreciate any kind of help. Thank you.</p>
<hr>
<p>Here is my attempt so far
... | Jack D'Aurizio | 44,121 | <p>Factor $n$ as $p_1^{\alpha_1}\cdot\ldots\cdot p_k^{\alpha_k}$. Then every solution is associated with three vectors (the exponents in the factorizations of $x,y,z$) with non-negative integer components and sum given by $(\alpha_1,\ldots,\alpha_k)$. By <a href="https://en.wikipedia.org/wiki/Stars_and_bars_(combinator... |
204,612 | <blockquote>
<p>Is it possible to verify the following <code>lhs,rhs</code> involving the sums
are equal, with Mathematica?</p>
</blockquote>
<p>I can verify it for individual values of <span class="math-container">$d$</span> variable:</p>
<pre><code>ClearAll[d, q, h, eq1, eq2, x, lhs, rhs];
eq1[d_: d, q_: q, h_:... | JimB | 19,758 | <p>I know you want the expectation for any positive value of <span class="math-container">$\kappa$</span> but here is the resulting expectation for integer values of <span class="math-container">$\kappa$</span>:</p>
<pre><code>mean[κ_, b_, μ_, σ_, ρ_, ω_] := b Exp[μ + σ^2/2] *
Sum[Binomial[κ, i] ω^i Exp[i ρ (μ + (1 ... |
235,945 | <p>Hello please help me with these trig identities and double angles as I am not sure where I am going wrong but I keep getting the wrong answer </p>
<p>This is the problem
$$
\sin(\theta+30) = 2\cos(\theta)
$$
This is my one of my incorrect solutions</p>
<p>$$\sin(\theta +30) = 2\cos(\theta)$$
$$\sin(\theta)\cos(30)... | Will Jagy | 10,400 | <p>As $\cos^2 \theta + \sin^2 \theta = 1,$ I'm afraid
$$ \cos \theta = \pm \sqrt{1 - \sin^2 \theta} $$
which is useless for your purposes. So leave the cosine on you right-hand side as it is, your substitution is just wrong. </p>
<p>Alright, if you do it properly, you get a relationship between $\sin \theta$ and $\... |
235,945 | <p>Hello please help me with these trig identities and double angles as I am not sure where I am going wrong but I keep getting the wrong answer </p>
<p>This is the problem
$$
\sin(\theta+30) = 2\cos(\theta)
$$
This is my one of my incorrect solutions</p>
<p>$$\sin(\theta +30) = 2\cos(\theta)$$
$$\sin(\theta)\cos(30)... | josh | 11,815 | <p>Your addition identity is almost correct. You can't say $\cos(\theta) = 1 - \sin(\theta)$. However, you can say $\cos^2(\theta) = 1 - \sin^2(\theta)$ from the Pythagorean Identity. Just be aware that you CANNOT drop the squares in this equation by taking the square root of both sides. Now,
\begin{array}{ccc}
\sin... |
31,099 | <p>I was wondering what "anti-optimization" is about? Is it related to optimization? What topics does it cover? </p>
<p>All I can find out from Google is <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ4-42WP6GH-M&_user=10&_coverDate=07/31/2001&_rdoc=1&_fmt=high&_orig=gate... | John | 99,531 | <p>Anti-optimization is a buzzword for "worst case" and the "hybrid optimization/anti-optimization" is a buzzword for <a href="http://en.wikipedia.org/wiki/Maximin_%28decision_theory%29%22maximin%22" rel="nofollow">Maximin</a>.</p>
<p>It should be pointed out that the invention of such buzzwords does not represent an... |
1,984,178 | <p>I have a problem with the following exercise:</p>
<p>We have the operator $T: l^1 \to l^1$ given by</p>
<p>$$T(x_1,x_2,x_3,\dots)=\left(\left(1-\frac11\right)x_1, \left(1-\frac12\right)x_2, \dots\right)$$ for $(x_1,x_2,x_3,\dots)$ in $l^1$. Showing that this operator is bounded is easy, but I am really desperate... | Hermès | 127,149 | <p><strong>Hint:</strong> Define $X^n = (\underbrace{\frac{1}{n}, \dots ,\frac{1}{n}}_{n}, 0, 0, ...)$. Show that for every $\epsilon >0$, there exists $N$ such that $||X^N||_{l^1} > 1-\epsilon$, thus proving the statement.</p>
|
1,662,226 | <p>Find a sufficient statistic for $σ^2$ with $μ$ known, where $X_i$ is a random sample from $N(μ,σ^2)$</p>
<p>I was able to find a sufficient statistic for $μ$ with $σ^2$ known, but I'm stuck on finding one for $σ^2$ when $μ$ is known. Can anyone give me some help? </p>
<p>I was using the factorization method before... | egreg | 62,967 | <p>Just square:</p>
<p>$0.4^2=0.16$</p>
<p>$0.04^2=0.0016$</p>
<p>$0.2^2=0.04$</p>
<p>$0.02^2=0.0004$</p>
<p>$0.13^2=0.0169$</p>
<p>Can you choose?</p>
|
1,662,226 | <p>Find a sufficient statistic for $σ^2$ with $μ$ known, where $X_i$ is a random sample from $N(μ,σ^2)$</p>
<p>I was able to find a sufficient statistic for $μ$ with $σ^2$ known, but I'm stuck on finding one for $σ^2$ when $μ$ is known. Can anyone give me some help? </p>
<p>I was using the factorization method before... | colormegone | 71,645 | <p>If you rationalize the denominator of your ratio to get $ \ \frac{ \sqrt{10} }{25} \ $ , you can use the fact that $ \ \sqrt{10} \ $ is a little bigger than 3 to estimate that the number in question is a bit larger than $ \ \frac{3}{25} \ = \ 0.12 \ $ . No other choice but (E) is close to that.</p>
|
1,200,919 | <p>Let $x$ be the solution of the equation $x^x=2$. Is $x$ irrational? How to prove this?</p>
| Vincenzo Oliva | 170,489 | <p>Suppose $x$ is rational. Then there exist two integers $a,b$ such that $$\left(\frac{a}{b}\right)^{a/b}=2 \\ \frac{a}{b} = 2^{b/a}.$$ But that's impossible because the RHS is rational only for $a=1$, which actually makes it also integer, while with $a=1$ the LHS is non-integer for all $b>1$. Checking that $(a,b)=... |
1,821,927 | <p>Let $V = \big\{z: |z|<5,\text{Im}(z)>0 \big\}$. Let $f$ analytic in $V$, continuous in $\overline{V}$ and suppose $$\forall x \in \left[ -5,5\right]:\ f\left( x\right) \in \mathbb{R}$$
Show that $$\limsup_{n \rightarrow \infty} \root{n}\of{\frac{f^{(n)}(1)}{n!}} \le \frac{1}{4}$$
<br><br><br>
I tried expanding... | mercio | 17,445 | <p>Let $V'$ be the complex conjugate of $V$, $\gamma$ the path following the border of $V$ in a direct orientation, $\gamma'$ the path following the border of the $V'$ also in a direct orientation, and $W$ the open ball of radius $5$ centered at $0$ (so that $\overline W = \overline V \cup \overline {V'}$)</p>
<p>Then... |
3,450,581 | <p>I know convergence-preserving functions have been discussed a fair amount in the past; however, I was a looking at <a href="https://math.stackexchange.com/questions/1337042/sum-a-n-converges-iff-sum-fa-n-converges/1337057#1337057">another post</a>, and I saw the following result: if <span class="math-container">$f$<... | Cye Waldman | 424,641 | <p>According to <em>A Catalog of Special Plane Curves</em>, J. Dennis Lawrence, Dover, 1972, the equations for the epicycloid are</p>
<p><span class="math-container">$$x=m\cos t-b\cos mt/b\\
y=m\sin t-b\sin mt/b
$$</span></p>
<p>where <span class="math-container">$t\in[-\pi,\pi]$</span> and <span class="math-containe... |
206,318 | <p>For plots like the one shown below, what is the syntax for adding filling between particular lines and the axis, but only in the negative region:</p>
<p><a href="https://i.stack.imgur.com/X4TME.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/X4TME.png" alt="enter image description here"></a></p>
| Kuba | 5,478 | <pre><code>Want // ClearAll;
x = 111;
Want /: SetDelayed[lhs_, Want[s_String]] := ToExpression[
s,
InputForm,
Function[rhs, SetDelayed @@ Hold[lhs, rhs], HoldAll]
]
H[x_, y_] := Want["x^2+y^2"]
H[3, 4]
</code></pre>
<blockquote>
<p>25</p>
</blockquote>
|
4,347,308 | <p><em><strong>Definition:</strong></em></p>
<p>Let <span class="math-container">$(X,\mathscr{A},\mu)$</span> be a measurable space, an atom of the measure <span class="math-container">$\mu$</span> is a set <span class="math-container">$A \in\mathscr{A}$</span> with the property that
<span class="math-container">$\mu(A... | heropup | 118,193 | <p>Don't do more work than you need to. The better approach is to perform the polynomial long division first:</p>
<p><span class="math-container">$$\frac{1+x^3}{1+x^2} = x + \frac{1-x}{1+x^2}.$$</span> Now apply your series formula:</p>
<p><span class="math-container">$$\begin{align}
x + \frac{1-x}{1+x^2}
&= x + ... |
153,409 | <p>Would you please tell me whether there is any wrong on this problem? given that $g$ is continuous on $[0,\infty)\rightarrow \mathbb{R}$ satisfying $\int_{0}^{x^2(1+x)}g(t)dt=x \forall x\in [0,\infty)$ then I need to find what is $g(2)$?</p>
| JLA | 30,952 | <p>Note that
$$
\frac{d}{dx}\int_0^{x^2(1+x)}g(t)\,dt=g(x^2(1+x))(2x+3x^2)=\frac{d}{dx}x=1.
$$
So now solve for $g(2)$.</p>
|
716,498 | <p>My Algebraic Topology book says </p>
<blockquote>
<p>Let $\Bbb{R}^n$ denote Euclidean n-space. Then $\pi_1(\Bbb{R}^n,x_0)$ is the trivial subgroup (the group consisting of the identity alone).</p>
</blockquote>
<p>I wonder why that is. I can imagine infinite continuous "loops" in $\Bbb{R}^3$ that start and end a... | Bruno Stonek | 2,614 | <p>The problem with your last sentence is that $\pi_1(X,x_0)$ is not the set of loops based on $x_0$, but of <em>homotopy classes</em> of loops based on $x_0$.</p>
<p>Can you see why every loop in $\mathbb R^n$ based on $x_0\in \mathbb R^n$ is homotopic to the constant map based in $x_0$?</p>
|
323,109 | <p>Could someone help with the following integration:
$$\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$$</p>
<p>So far I have done the following, but I am stuck:</p>
<p>I denoted $ y=-\cos x $ then:
$$\begin{align*}&\int^{1}_{-1} \frac{\arccos(-y) \sin x}{1+y^2}\frac{\mathrm dy}{\sin x}\\&= \arccos(-1) \arctan 1+\a... | Community | -1 | <p>Let $$I = \int_0^{\pi} \dfrac{x \sin(x)}{1+\cos^2(x)} dx = \int_{-\pi/2}^{\pi/2} \dfrac{(x+\pi/2) \sin(x+\pi/2)}{1 + \cos^2(x+\pi/2)} dx = \int_{-\pi/2}^{\pi/2} \dfrac{(x+\pi/2) \cos(x)}{1 + \sin^2(x)} dx $$
Now
$$\int_{-\pi/2}^{\pi/2} \dfrac{(x+\pi/2) \cos(x)}{1 + \sin^2(x)} dx = \int_{-\pi/2}^{\pi/2} \underbrace{\... |
323,109 | <p>Could someone help with the following integration:
$$\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$$</p>
<p>So far I have done the following, but I am stuck:</p>
<p>I denoted $ y=-\cos x $ then:
$$\begin{align*}&\int^{1}_{-1} \frac{\arccos(-y) \sin x}{1+y^2}\frac{\mathrm dy}{\sin x}\\&= \arccos(-1) \arctan 1+\a... | L. F. | 56,837 | <p>$$I=\int_0^{\pi} \frac{-x\sin x}{1+\cos^2 x}\,dx=\int_0^{\pi} \frac{(x-\pi)\sin x}{1+\cos^2 x}dx\quad(x\to \pi-x)$$</p>
<p>$$\Rightarrow I=\frac{\pi}{2}\int_0^{\pi}\frac{-\sin x}{1+\cos^2 x}\,dx$$</p>
<p>Let $t=\cos x:$</p>
<p>$$I=\frac{\pi}{2}\int_{-1}^{1}-\frac{1}{1+t^2}\,dt=-\frac{\pi^2}{4}$$</p>
|
446,148 | <blockquote>
<p>Let $z_1, . . . , z_n$ and $w_1, . . . , w_n$ be complex numbers. Show
that $$|z_1w_1 + ··· + z_n w_n|^2 ≤ \sum ^n _{j=1} |z_j|^2 \sum ^n
_{j=1}|w_j|^2$$</p>
</blockquote>
<p>I basically tried to use the proof given for real numbers but I feel that something must be wrong. Could somebody look over ... | Emily | 31,475 | <p>Let $\omega = z_1w_1 + z_2w_2 + \cdots + z_nw_n$. In general, $|\omega|^2 = \omega \overline{\omega}$.</p>
<p>$$\omega \overline{\omega} = (z_1w_1+z_2w_2 + \cdots + z_nw_n)(\overline{z_1}\overline{w_1} + \cdots + \overline{z_n}\overline{w_n}) \\
= z_1\overline{z_1}w_1\overline{w_1} + z_1\overline{z_2}w_1\overline{w... |
446,148 | <blockquote>
<p>Let $z_1, . . . , z_n$ and $w_1, . . . , w_n$ be complex numbers. Show
that $$|z_1w_1 + ··· + z_n w_n|^2 ≤ \sum ^n _{j=1} |z_j|^2 \sum ^n
_{j=1}|w_j|^2$$</p>
</blockquote>
<p>I basically tried to use the proof given for real numbers but I feel that something must be wrong. Could somebody look over ... | Pedro | 23,350 | <p>You have a canonical inner product in $\Bbb C^n$ given by $x\cdot y=\displaystyle \sum_{i=1}^n x_i\overline{y_i}$. Note $x\cdot x=\displaystyle \sum_{i=1}^n x_i\overline{x_i}=\displaystyle \sum_{i=1}^n |x_i|^2$</p>
<p><em>Claim</em> Let $V$ be a $\Bbb C$-vector space, and $\langle \cdot,\cdot\rangle$ an inner produ... |
3,845,570 | <p>Premises: <span class="math-container">$\neg(A \to B)\ ,\ \neg B \to C$</span> .</p>
<p>Conclusion: <span class="math-container">$C$</span></p>
<p>My intuition is that I should do a sub-derivation where I prove <span class="math-container">$\neg C$</span> is an absurdity. However, I soon run into issues. If I could... | Tortar | 704,856 | <p>Write the first premise as <span class="math-container">$\neg\neg(A \land \neg B) \equiv A \land \neg B $</span> , so <span class="math-container">$\neg B$</span> is true. Therefore, from the second premise it follows <span class="math-container">$C$</span>.</p>
|
3,845,570 | <p>Premises: <span class="math-container">$\neg(A \to B)\ ,\ \neg B \to C$</span> .</p>
<p>Conclusion: <span class="math-container">$C$</span></p>
<p>My intuition is that I should do a sub-derivation where I prove <span class="math-container">$\neg C$</span> is an absurdity. However, I soon run into issues. If I could... | Graham Kemp | 135,106 | <blockquote>
<p>My intuition is that I should do a sub-derivation where I prove <span class="math-container">$\neg C$</span> is an absurdity. However, I soon run into issues. If I could prove that <span class="math-container">$B$</span> is an absurdity, that would work also, but I'm not sure how to do so using the firs... |
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\... |
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | N. F. Taussig | 173,070 | <p>@expiTTp1z0 has addressed where you made your error. </p>
<p>I am going to show you how you can reduce the given problem a simpler one by using symmetry.</p>
<p>You wish to find the number of solutions of the equation
$$x_1 + x_2 + x_3 + x_4 = 22 \tag{1}$$
in positive integers not exceeding $6$. Since $x_k$, $1 ... |
271,824 | <p>I have a list= {4, 8, 10, 11, 12, 14, 16, 7, 9}</p>
<p>How can i partition the list by group of Arithmetic Progression with common difference 1 :</p>
<p>{{4}, {8}, { 10, 11, 12}, {14}, {16}, {7}, {9}}</p>
| bmf | 85,558 | <pre><code>(li1 /. Thread[
Rule[Complement[Union[li1, li2], Intersection[li1, li2]], x]])
</code></pre>
<blockquote>
<p><code>{x, {2, 9}, x, {4, 7}, {5, 6}, x, x, x, {9, 2}, x}</code></p>
</blockquote>
|
946,973 | <p>After completing the square, what are the solutions to the quadratic equation below?
<span class="math-container">$$x^2 + 2x = 25$$</span></p>
<p><img src="https://i.stack.imgur.com/AoFhV.png" alt="enter image description here" /></p>
<p>Honstely I think it's B. But I'm not sure.</p>
| Jasser | 170,011 | <p>You can verify your answrt using this.
The solution to general quadratic equation $ax^2+bx+c=0$ is given by the formula $\frac {-b+\sqrt {b^2-4ac}}{2a}, \frac {-b-\sqrt {b^2-4ac}}{2a}$
The given equation is $x^2-2x-25=0$
There fore after applying this formula you will get option c as your answer.</p>
|
3,154,244 | <p>I tried to ask this in a different way and did not correctly explain myself.</p>
<p>I am ok integrating the line <span class="math-container">$y = x$</span> , let us say from <span class="math-container">$0$</span> to <span class="math-container">$2$</span> using calculus.
If I want to get the square I can easily m... | SNEHIL SANYAL | 636,469 | <p>Consider a rectangle formed by the equation <span class="math-container">$y=K$</span> extending from <span class="math-container">$x=a$</span> to <span class="math-container">$x=b$</span>.
<a href="https://i.stack.imgur.com/OUM4D.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OUM4D.png" alt="ent... |
1,158,956 | <p>To show that orthogonal complement of a set A is closed.</p>
<p>My try: I first show that the inner product is a continuous map. Let $X$ be an inner product space. For all $x_1,x_2,y_1,y_2 \in X$, by Cauchy-Schwarz inequality we get,
$$|\langle x_1,y_1\rangle - \langle x_2,y_2\rangle| = |\langle x_1- x_2,y_1\rangle... | Sam Wong | 507,382 | <p>Let <span class="math-container">$\{y_n\}_{n=1}^\infty \in A^\perp$</span> s.t. <span class="math-container">$y_n \to y.$</span></p>
<p>Then <span class="math-container">$\forall x\in A,$</span> we have <span class="math-container">$$\langle y , x\rangle=\lim_{n\to\infty} \langle y_n , x\rangle=\lim_{n\to\infty}0=0... |
2,166,075 | <blockquote>
<p>Prove $a_1+\cdots+a_n=\dfrac{(a_1+a_n)n}{2}$ inductively.</p>
</blockquote>
<p>Where $a_i=a_{i+1}-r$.</p>
<p>I tried to start proving it inductively, but any try lead to a bad conclusion, so I ended up proving it by making $a_n$ depend on $a_i$.</p>
<p>But I didn't know how to prove it inductively,... | Tsemo Aristide | 280,301 | <p>$a_1+a_2={2{a_1+a_2\over2}}$. You have $a_2=a_1+r$ and recursively $a_n=a_1+(n-1)r$, $a_1+...+a_n=a_1+r+a_1+2r+...+a_1+(n-1)r=na_1+r{n(n-1)\over 2}$ =${{na_1+na_1+rn(n-1)}\over 2}$ =$n{{a_1+a_1r(n-1)}\over 2}=n{{a_1+a_n}\over 2}.$</p>
|
418,647 | <p>Sorry if the question is dumb. I am trying to learn representation theory of finite groups from J.P.Serre's book by myself. In section 2.6 on canonical decomposition, he says that let V be a representation of a finite group G, $W_1,...,W_h$ be the distinct irreducible representations of G, and let V = $U_1 \oplus ..... | Jared | 65,034 | <p>If you allow loops and/or multiple edges between vertices, then such a graph exists. Take $1$ vertex with $17$ loops, or two vertices with $17$ edges between them, and let the other vertices be isolated.</p>
<p>Now assuming we are working with a simple graph (no loops, and no multiple edges), then no such graph ex... |
2,965,989 | <p>Why <span class="math-container">$$p(y)=\int_0^\infty x\delta (y-x)dx=y\ \ ?$$</span></p>
<p>For me, <span class="math-container">$$p(y)=\int_0^\infty x\delta (y-x)dx=\int_{\{y\}}xdx=0.$$</span></p>
<p>If it would be written <span class="math-container">$\int_0^\infty xd\delta _y$</span>, then I would be agree wit... | GEdgar | 442 | <p>Mathematical approach. The <a href="https://en.wikipedia.org/wiki/Distribution_(mathematics)" rel="nofollow noreferrer">Schwartz distribution</a> <span class="math-container">$\delta(y-x)\;dx$</span> is a <a href="https://en.wikipedia.org/wiki/Measure_(mathematics)" rel="nofollow noreferrer">measure</a>. It is the... |
3,700,299 | <p>I want to show that <span class="math-container">$\int\limits_{-\infty}^\infty e^{-\pi x^2}dx = 1$</span>.</p>
<p>By definition
<span class="math-container">$$\int\limits_{-\infty}^\infty e^{-\pi x^2}dx = \lim\limits_{t\to\infty}\int\limits_{-t}^t e^{-\pi x^2}dx$$</span>
and since the integrand <span class="math-co... | Harish Chandra Rajpoot | 210,295 | <p>Let <span class="math-container">$\pi x^2=t\implies dx=\dfrac{t^{-1/2}dt}{2\sqrt{\pi}}$</span>
<span class="math-container">$$\int_{-\infty}^{\infty}e^{-\pi x^2} dx=2\int_{0}^{\infty}e^{-\pi x^2} dx$$</span>
<span class="math-container">$$=2\int_{0}^{\infty}e^{-t}\dfrac{t^{-1/2}dt}{2\sqrt{\pi}}$$</span>
<span class=... |
3,679,386 | <p>defining matrix exponentiation for natural numbers by repeated multiplication and defining it for <span class="math-container">$\frac{1}{n}$</span> by: <span class="math-container">$A^{\frac{1}{n}}$</span> is the matrix s.t. <span class="math-container">$(A^{\frac{1}{n}})^n=A$</span>.
for a rational number <span cla... | M. Wang | 549,229 | <p>If <span class="math-container">$\sum_{n=1}^\infty z_n$</span> converges then the sequence <span class="math-container">$a_m = \sum_{n=1}^m z_n$</span> is convergent. So in particular, it is a Cauchy sequence, which means that for all <span class="math-container">$\epsilon > 0$</span> there exists some <span clas... |
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