Convergence of sequences and different modes of convergence.

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Heat Ecuation : Exchange partial derviate and series

I'am having a problem when checking the validity of the solution i found for the heat ecuation: \begin{cases} U_{t}(x,t)=U_{xx}(x,t),\ {(x,t)\in (0,1)\times(0,+\infty)} \\ U(x,0) = x^2 - x\\U(0,t)=0\...
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1answer
19 views

Integral Convergence with parameters

I am finding it hard to approach this question: $$\int_0^{\pi/2} {1-\cos(x)^a\over x^b}\, dx$$ and I need to determine for which positive values of $a,b$ the integral converges. Thanks,
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1answer
36 views

Complex convergence of sequence Proof check and help

Prove that, for $z \in C$ , the sequence $(z^{n})$ converges if and only if $|z| < 1$ or $z=1$ Proof. Say $z^n$ converges to some $a$ then there exists $n_o$ such that for all $n\geq n_o$ $|z^n-...
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Convergence of expectations of a sequence of exponential random variables.

Suppose $\{X_n\}$ is a sequence of exponentially distributed random variables, where $X_n$ has mean $1/\lambda_n$. Suppose that $\lim_{n\to\infty}\lambda_n = \lambda>0$. Let $X$ be exponentially ...
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how to calculate infimum of Augmented Lagrangian?

should any body explain that how do we calculate these step? \begin{align*} L(x,y) &= f(x) + y^T(Ax-b)\\ g(y) &= \inf_x \, L(x,y) \\ &= \inf_x \, f(x) - \langle -A^Ty, x \rangle - \langle ...
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0answers
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On $\sum_{\substack{\zeta(\frac{1}{2}+i\gamma)=0\\0<\gamma<T}}\prod_{n=1}^\infty \left| 1-\frac{(\gamma\log x)^2}{n^2\pi^2}\right|$ as $O(\log x)$

On assumption that the identity (2) for a representation of $\pi(x)$ holds, see here Two Representations of the Prime Counting Function in this site Mathematics Stack Exchange, and since using the ...
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3answers
55 views

Boundedness and convergence of $x_{n+1} = x_n ^2-x_n +1$

Suppose that $x_0 = \alpha \in \mathbb{R}$ and $x_{n+1} = x_n ^2-x_n +1$. I am asked to study the boundedness of $(x_n)$ and then asked if $(x_n)$ converges. How can I show that $(x_n)$ is bounded? ...
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0answers
7 views

What's about of an analogous Riemann's function $R(X)$ for twin primes?

It is well know the so-called Riemann's explicit formula for the prime counting function $\pi(x)$ involving the density $J(x)$ for prime powers and how by Möbius inversion one recovers $\pi(x)$ and ...
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3answers
156 views

What IS conditional convergence?

I've gone through countless websites that promise answering "What is conditional convergence?" and instead give me "This is how you find if something is conditionally convergent". While it is all ...
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2answers
168 views

Value of this convergent series: $\frac{1}{3!}+\frac2{5!}+\frac3{7!}+\frac{4}{9!}+\cdots$

What is the value of- $$\frac{1}{3!}+\frac2{5!}+\frac3{7!}+\frac{4}{9!}+\cdots$$ I wrote it as general term $\sum\frac{n}{(2n+1)!}$. As the series converges it should be telescopic (my thought). But ...
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1answer
47 views

Convergence of steepest gradient descent

The description of gradient descent in Wikipedia says: $$x_{n+1} = x_n - \gamma_n\nabla F(x_n)$$ for $n = 0,1,2,...$ Suppose that $x_n$ converges to $x$. Then, is it always true that $\nabla F(x) = ...
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1answer
44 views

Proof that $ \sum_{i=1}^\infty a_n$ is converges almost surely.

Let $\{a_n\} $ be a positive number sequence and $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. If we have $$\sup\limits_{t>0} \left( t. \mathbb{P} \left\{ \sum_{i=1}^\infty a_n >t \...
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1answer
48 views

Show that $\frac{S_n}{n}\to 0$ in probability if $s<\frac{1}{2}$

Let $s\in\mathbb{R}$ and $X_1,X_2,\dots$ be independent random variables and with distributions: $$P(X_n=n^s)=P(X_n=-n^s)=\frac{1}{2}$$ Let $S_n=X_1+\dots+X_n$. Show that $$\frac{S_n}{n}\to 0 \text{ ...
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2answers
49 views

Limit and convergence/divergence of an integral

I was working on a problem concerning the function $$f(x) = \frac{x^2}{\ln(x)^\sqrt{x}}$$ asking for the value of $$\lim_{x \to \infty}f(x)$$ and for the convergence/divergence of $$\int_2^\infty f(x) ...
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0answers
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Nesterov's bound between quadratic and strongly convex cases?

Are there some examples of simple & strongly convex functions for which the convergence bound of Nesterov’s Accelerated Gradient Method is better than Nesterov’s bound for strongly convex case $\...
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1answer
21 views

Need know all ways to show function is continuous, convergent and differentiable [on hold]

Please tell me all ways to show / proof that a function is continuous, convergent and differentiable. continuous: show that function is differentiable if yes then it is continuous also convergent: ...
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4answers
55 views

Proving convergence or divergence of series: Tips and Tricks

I currently write an article where I collect some tips for students for proving the convergence or divergence of series. What tips and tricks do you know or use or teach? Remark: I will add some ...
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0answers
31 views

Proving that this recursively defined sequence converges.

The sequence is defined as such, with $a_1=1$, $$ a_{n+1} = \begin{cases} a_n + 1/n, & \mbox{if } a_n^2 \leq 2 \\ a_n - 1/n, & \mbox{if } a_n^2 > 2 \\ \end{cases}. $$ In the book, P.M. ...
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1answer
24 views

Local normal convergence equivalent to compact normal convergence

Let $X$ be an open subset of $\mathbb{R}^m$ and let $f_n\colon X\to \mathbb{C}$ be complex-valued functions. Then one has the following two notions: $\textbf{1.}$ The series $\sum\limits_{n=0}^{\...
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3answers
83 views

Will the expression $\sum_{i=1}^{n}{\frac{i^{2}}{n^{2}}}$ converge as n approches infinity?

I have the following expression: $$\lim_{n \to\infty}\ \sum_{i=1}^{n}{(\frac{i}{n})^{2}}$$ I am not quite sure whether it will converge or diverge. Can somebody tell me how to figure it out?
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5answers
62 views

Does this expression diverge or converge?

I have the following expression: $$\lim_{n \to \infty} \frac{2}{n^2} \ {\sum_{i=1}^{n}{\sqrt{n^2 - i^2}}} \ $$ I am not quite sure whether it will converge or diverge. Can somebody tell me how to ...
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1answer
34 views

Non-negative, integrable random variables which converge in probability and whose expected values have a finite limit

Suppose we have a sequence $X_1, X_2,...$ of non-negative, real random variables (not necessarily increasing) in $L^1$ which converge in probability to an integrable, non-negative random variable $X \...
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2answers
42 views

I need help understanding The Integral Test for series

For the following Series I have to show that the series qualifies for The Integral Test, then use it to determine if the series converges or diverges. here's my work where I apply the Integral Test, ...
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0answers
21 views

Where were my mistakes when I've combined $\eta(s)=\left(1-2^{1-s}\right)\zeta(s)$ and the Fourier series for the fractional part?

Let $s=x+it$ the complex variable, thus we are denoting $\Re s=x$. Combining the identity $$\eta(s)=\left(1-2^{1-s}\right)\zeta(s),$$ that holds for $0<x<1$, where $\zeta(s)$ is the Riemann Zeta ...
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0answers
15 views

Over sequential spaces and $B(H)$

We say that a topological space $X$ is sequential if the following holds : If $U$ is sequentially open then $U$ is open. By sequentially open we mean that $x \in U$ and $x_n \to x$ implies that $x_n$ ...
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3answers
77 views

Does the sequence $T_n=2^{-n}$ converge?

$$T_n=2^{-n}$$ How can I tell if this converges? with previous questions I have just let $n = \infty$, however I'm unsure about this one.
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1answer
699 views

How to prove convergence in mean implies uniform integrability?

My class notes and wikipedia both say that $X_n \xrightarrow{L^1} X$ $\Leftrightarrow \; X_n \xrightarrow{P} X$ and $X_n$ are uniformly integrable. I am trying to work through the proof. I am able ...
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1answer
44 views

$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$, for complex variable $z$.

I want to find this limit for complex variable $z$ $$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$$ In the real case I know $\sin(z)$ is bounded by $-1, 1,$ and the limit is $0$. But in the complex case ...
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6answers
946 views

Does the improper integral $\int_0^\infty\sin(x)\sin(x^2)\,\mathrm dx$ converge

Does the following improper integral converge? $$\lim_{B \to \infty}\int_0^B\sin(x)\sin(x^2)\,\mathrm dx$$
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Does the series $1-\frac12+\frac12-\frac1{2^2}+\frac13-\frac1{2^3}+\frac14-\frac1{2^4}+\frac15-\frac1{2^5}+\cdots$ converge or diverge?

$1-\frac{1}{2}+\frac{1}{2}-\frac{1}{2^2}+\frac{1}{3}-\frac{1}{2^3}+\frac{1}{4}-\frac{1}{2^4}+\frac{1}{5}-\frac{1}{2^5}+\cdots$ I've be trying to figure out how to write this series symbolically so I ...
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69 views

Show that linear functional $L(f) = \int_0^1 f(x) dx$ is continuous

Let $(C[0,1], d_1)$ be a metric space of all continuous functions $f:[0,1] \to \mathbb{R}$, $d_1$ is the $L_1$ metric $$d_1(f,g) = \int\limits_0^1 |f(x) - g(x)| dx$$ Show that linear functional $L(...
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1answer
56 views

Convergence of $\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$

The problem I'm facing is as it follow: For which values of $a$ the integral converges: $$\int_{0}^{1} \frac{\sqrt {e^2+x^2} - e^{\cos x}}{\tan^ax}dx$$ So far I figured that if $a< 1$, the ...
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3answers
168 views

Convergence of $\sum ( \cos \sqrt[3]{n^3 + \sqrt n + 7} - \cos \sqrt[3]{n^3 - 2\sqrt n + 3})$

I have some problem with this example: $$\displaystyle \sum_{n=2}^{\infty}\Bigg(\cos\Big(\sqrt[3]{n^3+\sqrt{n}+7}\Big) -\cos\Big(\sqrt[3]{n^3-2\sqrt{n}+3}\Big)\Bigg)$$ the only idea that crossed my ...
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1answer
51 views

Can you prove the convergence of $ \int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx $?

Can you prove the following improper integral is convergent? $$ \int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx. $$
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7answers
124 views

Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$

Problem statement: Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty $. , $n\in \mathbb{N}$ My progress LHS is ...
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Rate of convergence vs number of iteration

Can anyone explain to me the difference between rate of convergence and number of iterations for a numerical algorithm? Is it correct to say rate of convergence measure how fast the sequence approach ...
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1answer
77 views

Behavior of fundamental solution to heat equation after projection

I am considering the behavior of $$\frac{1}{h}\|(1-P_h)S(h)v\|,\tag{1}$$ and $$\frac{1}{h}\|(1-P_h)S(h)P_hv\|,\tag{2}$$ as $h\to 0^+$ for a fixed good enough $v$. I hope to show one of them converges ...
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41 views

Show that $\sum (-1)^n x^{(2^n)}$ has no limit as $x \uparrow 1$

Show that the following limit does not exist: $$\sum_{0}^{\infty} (-1)^n x^{(2^n)}\text{ with }x \uparrow 1$$ I tried setting $$f(x) = x - x^2 + x^4 - x^8...$$ then $$f(x) = x-f(x^2)$$ then the ...
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1answer
57 views

How to prove that if $E[X^2]$ is finite then $n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0$?

Let $X$ be a random variable with $E[X^2]<\infty$. I want to prove that $$ n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0 \text. $$ I tried to apply Chebyshev's inequality, ...
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17answers
14k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
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1answer
100 views

Prove that $\displaystyle\int_0^\infty \frac{\sin x}{x}dx$ converges using power series

$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-...=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!}$$ $$\frac{sin x}{x} = 1-\frac{x^2}{3!}+\frac{x^4}{5!}-...=\displaystyle\sum_{n=0}^\infty(-1)^...
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1answer
28 views

Pointwise convergence: State $f(x) = \lim f_n(x)$

I'm completely confused by this subject and hoping you guys can help me to clear up my confusion. So I'm told: State $f(x) = \lim f_n(x)$ where $f_n(x)=\frac{x^n}{\sqrt{3n}}$ for $x \in [0,1]$ ...
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2answers
199 views

If $K$ is compact and $C$ is closed in $\mathbb{R}^k$, prove that $K + C$ is closed using a “direct” proof

Rudin Exercise 4.25(a) reads: If $K$ is compact and $C$ is closed in $\mathbb{R}^k$, prove that $K + C$ is closed. The hints in the problem suggest a proof by proving that the complement of $K + C$ ...
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2answers
57 views

When a Markov chain converges to a steady state, what kind of convergence is it?

Let $A$ be a transition matrix, the steady state distribution $x$ satisfies the distribution $Ax = x$. One can prove that under certain circumstances, $$\lim_{n\rightarrow\infty}A^n q=x$$ where $q$ is ...
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3answers
286 views

Prove $\cos(n)$ does not converge as $n$ tends to infinity

How do I go about proving that $\lim\limits_{n \to \infty} \cos(n)$ does not exist where $n\in \mathbb{N}$ using an $\epsilon-N$ style method?
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2answers
43 views

On the convergence of a series $\sum_{n=1}^\infty \left( \frac { p(p+1) \cdots (p+n-1) }{ q(q+1) \cdots (q+n-1)} \right)^n$

I am struggling with the series $$\sum_{n=1}^\infty \left( \frac { p(p+1) \cdots (p+n-1) }{ q(q+1) \cdots (q+n-1)} \right)^n,$$ where $p,q>0$. I have checked the Dirichlet and ratio test so far, ...
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1answer
21 views

Limit of a function with exponential function and two parameters tending to infinity

I need some help with calculation of limits. I have a function $n(e^{it/\sqrt{m+n}}-1) + m(e^{-it/\sqrt{m+n}}-1) + \frac{m-n}{\sqrt{m+n}}it$ The solution says this converges to $-\frac{1}{2}t^2 \...
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1answer
68 views

For which $\alpha$, $\beta$ does $\int\limits_1^{\infty} x^{\alpha} \cdot (\ln x)^\beta dx$ converge? [duplicate]

For which $\alpha$ and $\beta$ does the following integral converge ?: $$ \int_{1}^{\infty}x^{\alpha}\,\ln^{\beta}\left(x\right)\,\mathrm{d}x $$ Here is my analysis: I noticed that the function ...
3
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2answers
77 views

Problem concerning the sequence $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$

The question is : Prove that the sequence $\{s_n\}$ where $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$ is convergent. Hence find $\lim_{n \to \infty} \left(1 - 1/2 + 1/3 - ... - 1/2n\right)$. I have ...
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0answers
12 views

Why do they add addition $\mathbb{E} (s_1^-)<\infty$ when they have $\{ s_n, \mathcal{F}_n,n\ge 1\}$ be a submartingale?

By definition, if $\{ s_n, \mathcal{F}_n,n\ge 1\}$ is a submartingale then $\mathbb{E} (|s_n|)<\infty$ for all $n\in \mathbb{N}^*$. But according to a arcticle, in a theorem, author add addition: $\...