Convergence of sequences and different modes of convergence.

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Testing convergence of series $\sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$ [duplicate]

Considering $$\sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$$ depending on $k$, which can be real. I have absolutely no clue how to proceed. Tried to taylor it, but with no result.
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1answer
36 views

Testing convergence of a series $\sum_{n=3}^\infty\ln\left(\frac{\cosh(\pi/n)}{\cos(\pi/n)}\right)$

Given series $$\sum_{n=3}^\infty\ln\left(\frac{\cosh(\pi/n)}{\cos(\pi/n)}\right)$$ I should test convergence. I know, that I should use a comparison criterion. I tried expressing $$2\cosh(x)=e^x+e^{-x}...
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1answer
28 views

radius of convergence of ${3^{k^2}}{x^{k^2}}$

find the interval of convergence $\Sigma _{k=0} ^{\infty} 3^{k^{2}} x^{k^{2}}$ The radius of convergence of this series is 1/3 by the book. but the answer and what i think is so different what i ...
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1answer
37 views

Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$?

Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$ I have a proof for the first case, under the assumption that $f$ is $C^1$ and real valued (also $1$ periodic) $\lVert ...
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3answers
36 views

Testing convergence of series $\sum_{n=2}^\infty\frac{\ln\left(\frac{n+1}{n-1}\right)}{\sqrt{n}}$

Lets have a series $$\sum_{n=2}^\infty\frac{\ln\left(\frac{n+1}{n-1}\right)}{\sqrt{n}}$$ However, I have absolutely no clue how to try to continue. I could probably use the integral criterion and ...
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0answers
28 views

Convergence of $\sum_{n=1}^{\infty}\log(1-e^{2\pi inz})$

Let $f(z)=\sum_{n=1}^{\infty}\log(1-e^{2\pi inz})$ be given on the upper half plane $H=\{z\in\mathbb{C}:\Im(z)>0\}$. Why does this function converges absolutely and uniformly on compact subsets of $...
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Branch points of power series and radius of convergence

The following text trys to give an answer to the following problem: Let $t(q)=\sum_{n=0}^{\infty}t_n q^n$ be a powerseries with radius of convergence $1$ and $t_0=0,~t_1\neq 0$. $w(q)$ is another ...
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1answer
16 views

Convergence of factor

In my math course there are some examples to test convergence of power-series with d'alembert. One of the examples is : $\sum_{n=0}^\infty \frac{x^n}{n!}$ Now i tried to solve this with d'...
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13 views

Möbius inversion for Taylor series and simple computations related with $\sum_{k=4}^\infty\frac{\mu(k)}{k^s}$ for $\Re s>\frac{1}{2}$

I've known an example of Möbius inversion for Taylor series (see the reference in my previous recent post, Benito, Navas and Varona, Möbius inversion from the point of view of arithmetical semigroup ...
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Möbius inversion for Taylor series and the logarithmic integral

See the first paragraph in page 77 of this article from Universidad de la Rioja (in english )Benito, Navas and Varona, Möbius inversion from the point of view of arithmetical semigroup flows (...
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1answer
19 views

Pointwise Convergence Uniqueness

I was reviewing my text from Real Analysis, and something occured to me that hadn't before, nor is it mentioned in the text. The way you usually show that a sequence $\{f_n\}$ of functions does not ...
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3answers
49 views

Prove $\sum_{k = 2}^\infty \ln(1+\frac{1}{k^2})$ converges using $\exp(x) \geq 1+x$.

All I've got so far is $$\exp(x) \geq 1+x \Rightarrow x \geq \ln(1+x) \Rightarrow \frac{1}{k^2} \geq \ln\left(1+\frac{1}{k^2}\right)$$ which (since $\ln(1+\frac{1}{k^2})$ is larger than zero) means ...
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1answer
37 views

Dominated convergence theorem exercise application

I have to prove that $$\lim_{n \to \infty} \int_0^\infty \mathrm (1+x/n)^{-n}(x^{-1/n})\mathrm{d}x = 1$$ I've been told to use Dominated convergence theorem but I can't find a function $|f_n(x)| \le ...
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19 views

Convergence of a series with terms in-between harmonic and geometric

Let $\alpha \in (0,1)$, and let $K \in \mathbb N$. Consider the infinite series as a function a $K$: $$f(K) = \frac{K^\alpha}{(1+K^\alpha)} + \frac{K^\alpha (K+1)^\alpha}{(1+K^\alpha)(1+(K+1)^\alpha)}...
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164 views

Convergence under a Hilbert space

Let $\{\varphi_n\}_{n=1}^\infty$ be an orthonormal sequence (not necessarily a basis) in a Hilbert space. Let $\{\lambda_n\}_{n=1}^\infty$ be a sequence of numbers Define $T:H\to H$ by $Tx= \sum_{n=...
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1answer
36 views

$L^2$ convergence of partial sum of a sequence of functions: $\sum_{k=1}^n\frac{g^k(x)}{k}$

Let $g:\mathbb{R}\to \mathbb{C}$ be an $L^2$ function such that $|g(x)|\leq\epsilon<1, $ for every $x\in\mathbb{R}$, and $g(x) = O\left(\frac{1}{x}\right)$. I want to know if $h_n(x):= \sum_{k=1}^n ...
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1answer
24 views

Uniform convergence of $\sum (-1)^nf_n(x)$ on $[0,1]$ where $f_n(x)=x^n(1-x)$.

Let $f_n(x) = x^n(1-x)$ and $\sum (-1)^nf_n(x)$. I showed that this series point wise. Case1) $x=1$ $$ f_n(1)=0 \, , \sum{(-1)^nf_n(1)} = 0 $$ Case2) $x\neq 1$ $$|f_n(x)| \leq |x|^n$$ since $$\sum |...
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1answer
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Does being Cauchy in $M$ imply being Cauchy in a subspace?

I've just seen an example about the sequence $x_n = \frac{1}{n}$ being Cauchy in $(0,1]$ because "Cauchy in $\mathbb{R}$ $\implies$ Cauchy in $(0,1]$". Is this true for a general metric space $M$ and ...
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55 views

Is there any proof for this convergence conjecture?

Let $\left\{u_i\right\}_{i=1}^m$ be a sequence of real vectors, i.e. $u_i\in R^n, 1\le i\le m $ such that $\sum_{i=1}^m u_i u_i^T$ is a positive definite matrix. Define: $$K_m:=\left(\sum_{i=1}^m u_i ...
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45 views

is there any convergent sub-sequence of a sequence of all rational numbers?

Let $(a_n)$ be a sequence of rational numbers, where all rational numbers are terms. (i.e. enumeration of rational numbers) Then, is there any convergent sub-sequence of $(a_n)$?
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1answer
24 views

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
28 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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3answers
37 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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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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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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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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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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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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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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73 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
73 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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2answers
71 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
25 views

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 ...
2
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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})\...