Convergence of sequences and different modes of convergence.

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Convergence of product of sequences of functions

Suppose we have two sequences of functions $(f^1_n),(f^2_n) $ where $f_n^1,f_n^2: \mathbb{R}^n \to \mathbb{R}$. These sequences verify ($\overset{u}{\rightarrow}$ means uniform convergence): $$f_n^1 ...
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1answer
27 views

If $\ d(x_n,x) $ exist then $\ (x_n) $ must be converge a point in $ X $ ?

Let $\ (X,d) $ be complete metric space, $\ x \in X $ and $\ (x_n) \subset X $ bounded sequence. If the real valued sequence {$\ d(x_n,x) $} convgergent then $\ (x_n) $ must be converge a point in ...
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36 views

For which $a>0$ does $\sum_{k=1}^{\infty} \frac{1}{k^a+a^{-k}}$ converge?

For which $a$ does $$\sum_{k=1}^{\infty} \frac{1}{k^a+a^{-k}}$$ converge? So far, I have figured out (and I hope I'm not wrong about this) that the series converges for $a > 1$ since $\frac{1}{k^...
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1answer
25 views

Convergence in metric space and show cauchy sequence in metric space

I consider a metric space ($C^1(\mathbb{R}), \mathbb{R}$), $\Vert \cdot \Vert_u$) where $\Vert \cdot \Vert_u$ is the uniform norm, and I want to show that for $f_n \in C^1$, that is the continous ...
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1answer
42 views

Calculus 2 - Prove Disprove - convergence of Taylor series

I got this question regarding properties of Taylor series. I'm stuck on the second question, I believe it is true since the area of convergence for X is affected by the coefficient and it is not ...
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39 views

About (0.999) equal to 1? [duplicate]

I saw the solution. $x=(0.999...)$ $\implies10x=(9.999...)$ $\implies10x-x=(9.999...)-(0.999...$) $\implies9x=9$ $\implies x=1$ Is this solution correct or wrong? Where does this solution lack?
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1answer
49 views

Exponential series property: $\alpha (z)=\sum_{n=-\infty }^{\infty }e^{-zn^{2}\pi}$

Good day, please a dude, show that if $$\alpha (z)=\sum_{n=-\infty }^{\infty }e^{-zn^{2}\pi}$$ then $\alpha(z^{-1})=z^{\frac{1}{2}}\alpha (z)$ for $\Re(z)>0$. I thought for properties of $e$, ...
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3answers
39 views

Asymptotic expansion of ratio function

I want to expand the following function: $$ f(x)=\frac{1}{(1-e^{-x})} $$ $f(x)$ can be rewritten as $$ f(x) \sim \frac{1}{x-x^2/2 + x^3/2/3} $$ But I want to express big-oh notation such that $$ f(...
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1answer
17 views

Does there exist a if and only if condition so that arc length of a convergent sequence of functions to converge to the arc length of the limit.

Is there a necessary and sufficient condition so that the arc length of a convergent sequence of functions converges to the arc length of the limit of the function? I know that if $f'_n$ is ...
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4answers
55 views

Convergence test for series $\sum_{n=2}^{\infty}\frac{(-1)^n}{\sqrt{n}\sqrt{n+(-1)^n}}$

What would be the approach to resolve whether this series converges (absolutely or conditionally) or diverges?
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0answers
37 views

Subspace of $L^1(\Omega)$ closed

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ two sub-sigma-algebras and $f$ a $\mathcal{B}$-measurable function. I want to show, that the subspace $$\...
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0answers
23 views

Does Cauchy condensation test relate to order of convergence/divergence?

For example harmonic series corresponds to Cauchy condensed series of $1+1+1+\cdots$ and since the Cauchy condensation is exponential, it just seems natural to reverse of it being related to order of ...
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55 views

Convergence of a process

this may be viewed as a duplicate of this post. However i have put in much effort in the shared link and donated it with reputation, to check the proof considered there. Here however i want to argue ...
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2answers
18 views

Sequence that converges point wise

How can I prove that $f_n(x)=x^n(1-x^n)$ with $0\leq x\leq 1$ converges pointwise to the function zero but not converges uniformly. I tried to get the maximum and then take the limit but I only ...
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0answers
11 views

Convergence and power means

If a non-negative sequence of numbers converges to some limit, does the n'th power mean of the sequence (constructed as a limit of finite power means on the first N entries) also converge to the same ...
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1answer
20 views

The convergence interval of two series's sum

I have the following statement: If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R_{1}$ and $\sum_{n=0}^{\infty}b_{n}x^n $ converges for $|x| < R_{2}$ , then $\sum_{n=0}^{\infty}(a_{n}+...
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1answer
36 views

Radius of convergence of $\sum_{k=0}^{\infty} c_n^2x^n$.

The problem statement is as follows: Suppose the radius of convergence of the complex valued series $\sum_{n=0}^{\infty} c_nz^n$ is $R$. Find the radius of convergence of $\sum_{n=0}^{\infty} c_n^2z^n$...
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0answers
24 views

Formal series which uniformly bounded in each representation of universal $C^*$ algebra converge

Let $A = C^*(T_1,T_2,...|T_i^* = T_i, ||T_i||\leqslant 1)$ - universal $C^*$ algebra of countable family of selfadjoint operators. I have formal series $x = \sum_{i_1,...,i_k} \alpha_{i_1, ..., i_k} ...
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Doubts and computations about Dirichlet series and aliquot sequences II

From previous post* dedicated to aliquot sequences I believe that I can state that for $\Re s>2$, on assumption that the Catalan-Dickson conjecture is false $$\sum_{n=1}^{\infty}\frac{s^{k+1}(n)-\...
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1answer
55 views

Does $\frac1n\sum\limits_{k=1}^na_k^2\to\rho$ with $0\le\rho<1$ imply$\prod\limits_{k=1}^na_k\to0$?

Let $\{a_n\}$ be a sequence of real numbers such that $\lim_{n\to \infty}\frac{\sum_{j=1}^n a_j^2}{n}=\rho$, and $0\le\rho<1$. The goal is to check whether the following is true $$\lim_{n\to \infty}...
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32 views

Showing that $\sum_{k=1}^{\infty}\frac{\ln k}{k^s}$ converges for all $\Re s > 1$

I want to prove that $\sum_{k=1}^{\infty}\ln k/k^s$ converges for all $\Re s > 1$. Since $\ln k/k^\alpha\to 0$ as $k\to\infty$ for all $\alpha > 0$, there exists a $N$ such that $\ln k<k^{\...
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63 views

What's the worst sequence that still leads to a converging series?

As a background, I'm initially interested in sequences $a_n$ giving rise to functions $\sum_{n=0}^\infty a_nx^n$ for $x\in(0,1)$ and their diverging behavior for $x$ to $1$. E.g. the geometric series $...
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3answers
72 views

Find radius of convergence and calculate sum $ \sum_{n = 0}^{\infty} \frac{x^{3n + 2}}{3n + 2} $

$$ \sum_{n = 0}^{\infty} \frac{x^{3n + 2}}{3n + 2} $$ For radius: I have tried to represent in the form of :$$ \sum_{n = 0}^{\infty} x ^n a_n $$ to use standart methods, but I failed. For sum ...
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1answer
39 views

Cocountable topology and limits of sequences

I need help solving the following statements: Look at the interval $[0,1]$ with the topology $$\mathcal T_c := \{M \subseteq [0,1]:M=\emptyset\quad or\quad [0,1] \backslash M \, is\, countable \}...
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0answers
28 views

On series of the kind $\sum_{n=1}^\infty\frac{1}{n^2}\cdot\frac{1}{(1+nx)^s}$ and Frullani's theorem

I would like to know if my computations, in this post are not required justifications for the computations unless if there is a mistake in my claims, were rights and solve a question as if are known a ...
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1answer
72 views

If the series $\sum_{k=1}^{\infty} a_k$ is Cesàro summable and $n a_n \to 0$ as $n \to \infty$, then the series converges

I'm learning about Fourier series, specifically Cesàro summable sequences and series, and need help with the following problem: Show that if the series $\sum_{k=1}^{\infty} a_k$ is Cesàro ...
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70 views

Prove or disprove: $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(-\infty,\infty)$

I'm trying to find out if $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(\infty,\infty)$. Here's my attempt at an answer: If $ x\in(-\infty,0]:$ $$e^{-|x-n|}=e^{(-(x-n))}=e^{x-n}\le e^{-n}$$ ...
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2answers
46 views

Does this series converge? (test for convergence)

I have a series defined like this: $$\sum_{n=1}^{\infty} (-1)^n (\cos \frac{1}{n})^n$$ and I need to find out whether it converges or diverges. I know that $\lim_{n\to\infty} |a_n| = 1$ but does it ...
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1answer
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Doubts concerning to an application of Frullani's theorem to $f_k(x)=\frac{2^{-k^2x}-2^{-(k^2+1)x}}{x}$, and Lebesgue convergence theorems

By application of Frullani's theorem for $a_n=n^2+1$, $b_n=n^2$ where $n\geq 2$ and $f(x)=2^{-x}$ then RHS in Frullani's integral is obtained for $n\geq 2$ as $$\log(1-\frac{1}{n^2}),$$ thus I asked ...
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1answer
15 views

Summation of L1-function values finite

For $f \in L^1(\mathbb{R})$, we can proof, that $\sum_{n\in \mathbb{Z}} |f(t+na)|<\infty$ ALMOST EVERYWHERE for $t\in [0,a]$ ($\star$ proof below) My Question: If we assume $f\in L^1(\mathbb{R})\...
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1answer
66 views

how can I find the convergence of the integral $\int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2}~dx$ , for $ x \in [0,1]$ [closed]

I want to check the convergence of the integral $$\int_{0}^{1}\frac{\ln(1+n^2x^2)}{n^2} dx $$ for $ x \in [0,1]$ and n->∞ is a constant so can basically pulled out of the integral but I don't know ...
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3answers
67 views

how can I find the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , $ x \in (-1,1)$ [closed]

I want to check the convergence of the integral $\displaystyle\int_{-1}^{1}\frac{1-x^n}{1-x}$ , for $ x \in (-1,1)$ but i don't know what to do. Every theory I know it is not working. Can someone ...
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1answer
27 views

Radius of Convergence on Power Series Help

I am struggling to find the radii of convergence of the following two series: $$\sum_{n}n^{\cos(n)}z^n$$ $$\sum_{n}(2^{-n} + 3^{-n})z^n$$ Here I tried using ratio test and lim sup, but didn't ...
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1answer
38 views

If $f: \mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic, then $f \in L^2[-\pi, \pi]$

I'm learning about Fourier series, specifically $L^2$ convergence, and need help with the following problem: Let $f: \mathbb R \to \mathbb R$ be continuous and $2\pi$-periodic. Show that $f \in L^...
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29 views

Is this a rigorous enough argument against uniform convergence?

I have $\sum_{n=1}^{\infty} \frac{ln(1+x/n)}{n}$ and I need to prove that it does not converge uniformly on $[0,\infty)$. So I use the delta-epsilon definition of convergence and then I expand this ...
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1answer
31 views

Find widest subset on which Fourier series can be integrated and derived term by term

As part of one problem I need to find the widest subset of $\mathbb{R}$ on which the obtained Fourier series can be integrated and derived term by term. I found that it has something to do with ...
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1answer
82 views

$\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ - Theory of distribution

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...
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0answers
41 views

Convergence of sequence of vectors in $C^*$-algebra

Let $B$ is $C^*$-algebra and $x_i \in B$ - linear independent vector system, $\alpha_i \in \mathbb{C}$ such that: $$\|x_i\| = 1$$ $$\lim_{N \to \infty} \|\sum_{k=1}^N \alpha_k x_k\| = \lim_{N \to \...
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0answers
46 views

Evaluating convergence of a non-trivial series

I am trying to evaluate whether the following limit is finite (as opposed to being $\infty$): $$\lim_{n \to \infty} \sum_{k=2}^n \frac{1}{n-1} \left \{\sum_{i=2}^l (i-1)\frac{(n-i)!}{(n-i-k+2)!} \...
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2answers
68 views

Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis

Let $A = C^*(T_1,T_2,T_3,... | T_i=T_i^*, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of self-adjoint operators. $A$ have standard Schauder basis, which contains all ...
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107 views

If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$

I have the following statement: If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$ as well. I couldn't find a ...
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3answers
76 views

Evaluate $\int_0^1y ( ( 1+\frac{1}{y^2} )\log (1+y^2) -1 )dy=-1+\frac{\pi^2}{24}+\log 2$ and a related generalization

Let $0<x<1$ and $0<y<1$ thus $\xi=xy^2<1$ and we can use the series expansion $$\frac{1}{2}\log\frac{1+\xi}{1-\xi}=\sum_{n=0}^\infty\frac{\xi^{2n+1}}{2n+1}$$ to get $$\frac{1}{2}\int_0^...
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1answer
21 views

Show that $\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$ is convergent for $\Re s > -1$

I am struggling to understand the example in Special Functions p. 621, that states, that $$\int_{1}^{\infty}\frac{x-\left \lfloor x \right \rfloor -\frac{1}{2}}{x^{s+1}}(1-s\log x)\,\mathrm{d}x$$ is ...
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1answer
15 views

$<f_a, \varphi> = \int_{-\infty}^{\infty} \frac{\varphi(x)}{|x|} dx$ know that $\varphi(0)=0$ - Generalized function

Question : Show that $<f_a, \varphi> = \int_{-\infty}^{\infty} \frac{\varphi(x)}{|x|} dx$ for any $\varphi \in D(\mathbb{R})$ for which $\varphi(0)=0$. I am a little bit confused how to solve ...
2
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1answer
29 views

$\lim_{n \to \infty} \langle f_n, \varphi \rangle$ - Generalized function

Question : Let $f_n$ be the distribution $<f_n,\varphi>=n(\varphi(\frac{1}{n})-\varphi(\frac{-1}{n}))$. What distribution is $\lim_{n \to \infty} <f_n, \varphi>$ ? First try : $\lim_{n \...
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1answer
34 views

The sum of series in an interval

I have the following series - $$ \sum_{n=1}^\infty nx^{2n-1} $$ I found that its convergence interval is $[-1,1]$ but how can i calculate the sum in this interval ? i would like to get some hint ...
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0answers
18 views

convergence of $\sum_{n=1}^{\infty}\frac{\sigma_k(n)}{n^s}=\zeta(s)\zeta(s-k)$

The identity $$\sum_{n=1}^{\infty}\frac{\sigma_k(n)}{n^s}=\zeta(s)\zeta(s-k)\qquad (1)$$ is well-known and valid for $s\in\mathbb{R}$ with $s>\max\{1,1+k\}$. $\sigma_k$ is the divisor function. ...
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0answers
15 views

Proof: Majorant- and minorantcriterion for convergence of improper integrals

Let $I$ a interval and let $f,g:I\to \mathbb{R}$ continuous with $0\le g(x) \le f(x) \forall x \in I$. Prove this propostitions: (a) If $\int_{0}^{\infty} f(x)dx$ convergent $\Rightarrow$ $\int_{0}^{\...
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1answer
67 views

Determining convergence of a series $\sum_n (-1)^n \sin a_n $

I need to determine if the following series is convergent: $$\sum_{n=2}^\infty (-1)^n\sin\left(\frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n\ln^2(n)}\right).$$ I've tried to use alternating series test but ...