Convergence of sequences and different modes of convergence.

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21 views

Completeness of a system: For all $n$ and within an interval?

Why is the system $\sin((2n-1)x)$ for $n=1,2,\cdots$ complete in $L^2[0,\frac\pi2]$? This means that the Euclidean norm converges for $n=1,2,\cdots$ and for all $x\in[0,\frac\pi2]$ How does one prove ...
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2answers
25 views

Convergent series? The pairwise product of the harmonic series with a nonnegative, diminishing to zero, divergent series

Suppose we have a sequence $\{\alpha_n\}$ such that $\alpha_n \ge 0$, $\sum_n \alpha_n = \infty$ and $\alpha_n \rightarrow 0$. Is it true that $\sum_n n^{-1} \, \alpha_n$ converges?
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52 views

How to prove that this series converges. [duplicate]

I want to prove that this series converges: \begin{equation} 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+... \end{equation} I normally use this one as a standard series to test other series for their ...
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1answer
31 views

Example of each $\{f_n\}$ Riemann integrable such that $\sum f_n$ converges point-wise to $f$ which is not Riemann-integrable

I am looking for an example of a sequence $\{f_n\}$ of real valued Riemann integrable functions on a closed bounded interval such that $\sum f_n$ converges point-wise to a function $f$ which is not ...
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5answers
49 views

Help with proof that $\sum_{n \in \Bbb{N}} \frac{1}{an + b}$ also diverges?

We know that $\sum_{n \in \Bbb{N}} \frac{1}{n}$ diverges. So it seems likely that $\sum_{n \in \Bbb{N}} \frac{1}{a n + b}$ will for any real $a, b$. I'm having trouble proving it just for the ...
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5answers
102 views

Prove that $\sum\frac{n+1}{(n+2)n!}$ converges

Show that $\displaystyle\sum\limits_{n=1}^{\infty}\frac{n+1}{(n+2)n!}$ converges, using the integral test. I noticed that $\displaystyle\sum\frac{n+1}{(n+2)n!} = \sum\frac{(n+1)^2}{(n+2)!}$, but ...
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796 views

Why does L'Hôpital's rule work for sequences?

Say, for the classic example, $\frac{\log(n)}{n}$, this sequence converges to zero, from applying L'Hôpital's rule. Why does it work in the discrete setting, when the rule is about differentiable ...
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2answers
33 views

locally uniform convergence vs pointwise convergence

I am finding lots on here about 'uniform convergence vs pointwise convergence' of a function but not the comparison for local uniform convergence. It somehow intuitively seems to me that pointwise ...
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3answers
60 views

Prove that $\sum\frac{(\log n)^2}{n^3}$ converges

This question is from Serge Lang's textbook, in a chapter that comes before the ratio and integral tests are introduced, so those can't be used. I've already proved that $\sum\frac{\log n}{n^3}$ ...
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2answers
51 views

Find $\sum_{n=1}^{\infty}a_nx^n$ given $a_0=3, \ 3na_n+3(n-1)a_{n-1}=2a_{n-1}$

Given $\ a_0=3$, $\,3na_n+3(n-1)a_{n-1}=2a_{n-1}$, find $\ f = \sum_{n=1}^{\infty}a_nx^n$. I have proved that when $\ \left\lvert x \right\rvert<1$, this exponential series function is convergent. ...
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1answer
55 views

To be or not to be Banach? That is the question.

On the set $H^1_0((0,2))$ we put the following norms. $$\|u\|_a^2= \int_{[0,2]}(u')^2.$$ $$\|u\|_b= \|u\|_\infty.$$ $$\|u\|_c= \|u\|_{L^2}.$$ Is $H^1_0((0,2))$ Banach with any of these norms?
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2answers
37 views

Counter Examples for Dominated Convergence Theorem and Fatou's Lemma

Is there an example to see why the dominated convergence theorem fails when there is no integrable function dominates the sequence $f_n(x)$? Also for Fatou's lemma, is there an example where the ...
0
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1answer
29 views

How to show that $L^p$ norm is monotone increasing?

I am trying to solve the following (very standard) exercise: Let $(X,\mathcal M,\mu)$ be a measure space and $f\in L^r\cap L^\infty$ for some $1\leqslant r<\infty$. Then $f\in L^p$ for ...
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2answers
24 views

Interval of convergence for a power series with $x^{2n}$

By definition, the radius of convergence (which is equivalent to the interval) is: $$R:=\frac{1}{\varlimsup_{n\rightarrow+\infty}\sqrt[n]{|a_n|}}$$ Where $\varlimsup_{n\rightarrow+\infty}$ is the ...
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2answers
44 views

How to integrate to find each function and its limit (if it exists)?

Find the expression for following three functions and then evaluate their limit. $$f(x,y) = \int\limits_y^x-\cos({\pi t})\ln{t}\quad dt\qquad\qquad x\gt y$$ $\mathbf 1.$ $$\lim\limits_{x\to\infty} ...
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20 views

Partial sum of a product over an arbitrary sequence.

Below is an equation a friend showed me, but was unable to prove. After struggling with it for a bit I was unable to as well. After failing to show this for, say, N=2 Im pretty sure the equation is ...
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1answer
44 views

Convergence of the integral $\int_0^1 \frac {1}{x\sqrt {1+x^\beta}}dx$

Is my integral-convergence contradiction proof valid? I have to brush up on my proof making. I am a little rusty. I was not sure if the following really held up. I wanted to prove the following is ...
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3answers
53 views

Will this series converge, will its limit exist (and if so, what is it), and will its sum be $> 0$ or $< 0$?

The series : $$\lim\limits_{n \to \infty } \sum\limits_{k=1}^n - (-1)^{k} \ln (k)$$ We know that, for a sum $\sum\limits_{k=1}^\infty a_k$ to converge, a necessary condition is that : ...
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2answers
54 views

For what $x$ does $ \sum_{n=1}^\infty x^n \sin (xn)$ converge?

Find the area of convergence for $$ \sum_{n=1}^\infty x^n \sin (xn)$$ Its easy to see that for $ x \in (-1,1)$ it converges, but not sure how to continue from there.
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2answers
31 views

Sum of $n=1$ to infinity of $\frac {\arctan\,n }{n^{\frac65}}$, converge or diverge?

http://www.wolframalpha.com/input/?i=Sum+of+n%3D1+to+infinity+of+tan+inverse+%28n%29+%2F+n%5E1.2 These were my thoughts on how to prove it. Since $\arctan\,n \gt \dfrac1n$ for $n\gt2$, we can say ...
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1answer
14 views

Exponential and convergence in $L^2$ bis

This question is a continuation of my question "Exponential and convergence in $L^2$" posted above: Let $(f_k)$ be a sequence of elements of $L^\infty(\Omega)$, which converge in $L^2(\Omega)$ to ...
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1answer
123 views

Does the sum converge for all values of $a$?

Here is the sum (for which $a$ is the variable): $$a+\sin(a)+\sin(\sin(a))+\cdots$$ Does the sum always converge for all values of $a$? So far, this is what I have done: 1) I plugged in many ...
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$\left\{x^k\right\}$ converges to $x^*$ superlinearly iff $\left\|\nabla^2f(x^k)^{-1}\nabla f(x^k)+x^{k+1}-x^*\right\|=o(\left\|x^{k+1}-x^*\right\|)$

Let $(x^k)_{k\in\mathbb N}\subseteq\mathbb R^n$ be convergent to $x^*$. We say, that the convergence is superlinear iff $$\left\|x^{k+1}-x^*\right\|=o\left(\left\|x^k-x^*\right\|\right)\tag{1}\;.$$ ...
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2answers
44 views

Convergence of Series and Infinite Sums [closed]

Should not all series converge to infinity....if you sum them over infinity?
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51 views

for what values of $a$ does $ \int_0^1 (-\ln{x})^a \,dx$ converge

for what values of $a$ does $$ \int_0^1 (-\ln{x})^a \,dx$$ converge? So far using the substitution $ t = -\ln{x}$ I got $\int_0^\infty t^ae^{-t} \,dt$ and using dirchelet it converges for every $ ...
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2answers
37 views

Computing $\Sigma_{n\ge3}^\infty\frac{n}{(n^2-n-1).\ln{n}.(\ln{(\ln{n})})^a}$

I've tried applying Raabe's, d'Alembert's and Cauchy's (root) convergence test but none of them get me very far. I also thought about trying Cauchy's integral convergence test but integrating this ...
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1answer
27 views

Show convergence by using comparison test $\sum_{n \geq 2} \dfrac{1}{n(n-1)}$

I am having problems with proving that $\sum_{n \geq 2} \dfrac{1}{n(n-1)}$ converges by using the comparison test. I have shown that it converges by finding its sum. It's a telescoping sum and its ...
2
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1answer
47 views

Understanding convergence in normed spaces and the language used when talking about norms.

We have the following definition about convergence in a normed space: "Let $(x_n)_{n=1}^\infty$ be a sequence in a normed space $(X,\|\cdot\|)$. We say that $x_n\to x$ in $X$ if, $$d(x_n,x)\equiv ...
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1answer
22 views

Can the Weierstrass M Test be used to find out where an Infinite Series Converges to?

I was under the impression that there was no analytic method to find out to what value an infinite series converges to - using methods like the ratio test one can find out if a series diverges or ...
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0answers
31 views

Sequence of functions that converge uniformly on a compact metric space.

I'm getting stuck with this problem: Let $\mathbb{X}$ be a compact metric space and $f_n, g_n: \mathbb{X} \rightarrow \mathbb{R}$ be functions converge uniformly to $f, g: \mathbb{X} \rightarrow ...
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1answer
94 views

$ \sum_{n=0}^\infty a_ne^{-\alpha_nx} $ converges for $x_0$ prove it uniformly converges in $[x_0,\infty]$

consider the folloing sum $$ \sum_{n=0}^\infty a_ne^{-\alpha_nx} $$ for every n $ 0 < \alpha_n < \alpha_{n+1}$ it is also given that for $x_0 \in \Bbb{R}$ the sum converges prove that the ...
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4answers
64 views

does $\int_0^\infty \frac{x-\arctan(x)}{x(1+x^2)\arctan(x)} \,dx$ converge

does the following integral converges? $\int_0^\infty \frac{x-\arctan(x)}{x(1+x^2)\arctan(x)} \,dx$ I calculated $$\int \frac{x-\arctan(x)}{x(1+x^2)\arctan(x)} \,dx = \ln(\arctan(x)) - \ln(x) + ...
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1answer
39 views

Exponential and convergence in $L^2$

Let $(f_k)$ be a sequence of elements of $L^\infty(\Omega)$, which converge in $L^2(\Omega)$ to $f\in L^2(\Omega)$. Where $\Omega $ is an open bounded subset of $R^n$. Is it true that : $e^{f_k} $ ...
2
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1answer
117 views

if $ L = \lim_{x\to1^-} \sum_{n=1}^\infty a_nx^n$ then $\sum_{n=1}^\infty a_n = L$

let ${a_n}$ be a sequence of Real non negative numbers. assume the following limit exists and is finite: $$ L = \lim_{x\to1^-} \sum_{n=1}^\infty a_nx^n$$ prove that $\sum_{n=1}^\infty a_n$ ...
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0answers
23 views

A problem upon function series

Function series $\sum_{n=1 }^{ \infty} u_{n}(x)$ converges to $S(x)$ in bounded interval $[a,b]$, if every $u_n(x)$ is non-negative and continuous in $[a,b]$. prove that $S(x)$ attains its infimum in ...
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Assume that $f_n\to f$ in measure and $\sup_n\|f_n\|_{L^p(E)}<\infty$ for some $p>1$. Prove that $f_n$ converges to $f$ in $L^1$ norm.

Let $\{f_n\}$ and $f$ be Lebesgue measurable functions on $E$ where $|E|<\infty$. Assume that $f_n\to f$ in measure and $\sup_n\|f_n\|_{L^p(E)}<\infty$ for some $p>1$. (a) Prove that ...
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91 views

If $\prod_{n}(1+a_n)$ converges, does $\sum_{n}\frac{a_n}{1+a_n}$ converge?

I have a sequence of complex numbers $a_1,a_2,...$ such that $a_i \neq -1$. I then have the infinite product $\prod_{n=1}^{\infty}(1+a_n)$ which I know converges to a non zero complex number. I was ...
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1answer
30 views

The Comparison test or The limit comparison test

I have checked many of this site pages yet I could not find a clear answer about how to choose between The "Comparison test" OR The "limit comparison test"? Because the difference between the two is ...
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Generalization of the Central Limit Theorem for Local Martingales

the Central Limit Theorem for Local Martingales states the following. Theorem Let $M_n = (M_n(s))_{s \geq 0}$ be a square integrable local martingale such that for all $T > 0$ $$ \lim_{n ...
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1answer
96 views

calculate $\prod_1^\infty k^{\frac1{k!}}$

Is it possible to find the point of convergence of $\prod_1^\infty k^{\frac1{k!}}$ $K!=k(k-1)!$. My attempt: If $S_n=\prod_1^\infty k^{\frac1{k!}}$ then $\ln S_n=\sum_1^\infty \frac{\ln k}{k!}< ...
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4answers
72 views

How to find the limit of a Sequence $ f(n) = \frac{16f(n-1)-2}{32} $

I have this sequence: \begin{equation} f(n) = \frac{16f(n-1)-2}{32} \end{equation} with \begin{equation} f(0) = 1 \end{equation} I want to find it's limit while n goes to infinity and therefore check ...
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0answers
19 views

How are the tolerances evaluated in fmincon? specific/complete mathematical formulations needed.

I'm currently studying the stopping criteria about fmincon using different algorithms and I'm wondering how are the tolerances are actually evaluated and compared in the built-in function fmincon. ...
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1answer
57 views

is $ f_n(x) = (1+x^n)^{1/n}$ uniformly convergent

does the following series of function is uniformly convergent in $[0,\infty]$? $ f_n(x) = (1+x^n)^{1/n}$ I found that $f_n \to f(x) = \begin{cases} x, & 1 \le x \\ 1, & 0 \le x \le 1 ...
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2answers
108 views

Computing an improper integral with respect to a parameter

I am motivated by this problem.Let us compute an improper integral with respect to a parameter:$$F(x)=\int_{1}^{\infty}\frac{e^{-xy}-1}{y^{3}}dy,\quad x\in[0,\infty).$$ The following is my ...
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0answers
71 views

Can I use Dominated Convergence Theorem for this sum?

$$\left[\sum_{n=0}^\infty \frac{a_n}{n!} (sz)^n\right]e^{-s}$$ Does this product decay to zero, as $s$ goes to infinity? Assume the left-hand term, call it $\phi(sz)$, is entire. Also, you can ...
0
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1answer
27 views

How to show that $P(X_{n} \geq n$ $i.o.) = 0$, given $E(X_{i}) = 0$ and $E((X_{i})^{2})=1$ for $i=1,2,3…$

I'm struggling with the following problem (Exercise 4.5.16 in Rosenthal's probability book): Let $X_{1}, X_{2},...$ be defined jointly on some probability space, with $E(X_{i}) = 0$ and ...
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2answers
46 views

How to show convergence in a $\mathbb{R}^n$?

I've come across a chapter in my book which has me stumped and nowhere can I find so that I can move on. The question is "Using the definition 9.1i, prove that the following limits exist: a) $$x_k = ...
0
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2answers
53 views

Convergence of a sequence by induction: $x_{n+1}=\frac{x_n+1}{x_n+a},x_1>0,a>0,n=1,2,…$

Assume that $x_n>0$ and prove $x_{n+1}>0$ $x_{n+1}=1-\frac{a-1}{x_n+a}$ $x_n+a>a$ $-\frac{a-1}{x_n+a}>-\frac{a-1}{a}\Rightarrow x_{n+1}>0$ Is it necessary to find upper bound to ...
0
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0answers
12 views

Convergence random matrix inverse

I have the following problem: A is a sum of independent random matrices that converges in expectation to say a matrix C and B is some fixed positive (semi-)definite matrix. I'm interested in a bound ...
4
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1answer
92 views

How do I evaluate this limit :$\displaystyle \lim_{x\to \infty} (1+\cos x)^\frac{1}{\cos x}$?

I would like to know if this :$$ \lim_{x\to \infty} (1+\cos x)^\frac{1}{\cos x}$$ does exist and how do i evaluate it ?. Note : I have tried to use the standard limit : $$ \lim_{z\to \infty} ...