Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

0
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0answers
29 views

$\sqrt{n}(x_n-x_0) \to N(0,\sigma_0^2)$ in distribution implies $x_n \to x_0$ almost surely

Is this true? I figured that if not, there will some positive probability $\sigma$ that $\sqrt{n}(x_n-x_0)$ takes $\sqrt{M} \cdot \epsilon$ for infinitely many large $M$. Even though this "blowing up" ...
1
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2answers
36 views

Find the range of convergence of the series$\,\,\sum_{n=0}^\infty {\frac{z^n}{1+z^{2n}}}$

The series I have is $$\displaystyle\sum_{n=0}^\infty {\dfrac{z^n}{1+z^{2n}}}$$ The same series with absolute values is: $$\displaystyle\sum_{n=0}^\infty {\dfrac{|z|^n}{1+|z|^{2n}}}$$ Using D'...
7
votes
2answers
322 views

Using the root test when the limit does not exist

I used the root test for the series $$ \sum_{n=1}^{\infty} \left(\frac{\cos n}{2}\right)^n. $$ I showed that $$ 0 \le \left|\frac{\cos(n)}{2}\right| \le \frac{1}{2} \implies \lim_{n\to\infty}\left|\...
0
votes
1answer
28 views

Absolutely convergent series of complex functions.

I have to do the following excercise: Let $\{f_n(z)\}_{n\in\mathbb{N}}$ a sequence of complex functions, and let $\sum_{n=1}^\infty f_n(z)$. Prove that: if $\sum_{n=1}^\infty |f_n(z)|$ converges, ...
1
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1answer
48 views

If $\sum g_n(x)$ converges uniformly and absolutely and $|f_n(x)|\leq |g_n(x)|$ show that $\sum f_n(x)$ converges uniformly and absolutely.

I do not know how to prove if the statement above is true. I know i can use the Cauchy criterion i.e. $|\sum_{n\rightarrow m}f_n(x)|\leq\sum_{n\rightarrow m}|f_n(x)|\leq \sum_{n\rightarrow m}|g_n(x)...
1
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2answers
67 views

Is absolute convergence a topological concept?

An infinite series $\Sigma_n a_n$ is said to absolutely converge if $\Sigma_n |a_n|$ converges. Absolute convergence implies convergence. My question is, is absolute convergence a topological ...
1
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1answer
26 views

Extending uniform convergence of analytic functions on larger domains

Let $f_k, f: ]-\infty , 1 [ \to \mathbb {R}$ be analytic functions. Suppose $f_k $ converges uniformly to $f $ on $]-\infty,0] $. Is it true that $f_k$ converges to $f$ on $]-\infty, \epsilon [$ for ...
2
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1answer
32 views

Example divergent series of analytic functions?

Does there exist a sequence of strictly increasing analytic positive functions $a_i : ]-1,1 [\to \mathbb{R}^{>0}$ such that $$f (x) = \sum_{i=0}^{+\infty} a_i (x) $$ converges for $x\leq 0$ and ...
0
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1answer
48 views

Absolute convergence of a power series. [closed]

Is it possible to have a power series with radius of convergence $R $ such that there exist $z_1$ and $z_2$ satisfying $|z_1|=R$, $|z_2|=R$ whereas the power series absolutely converges at $z_1$ and ...
1
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1answer
70 views

Examining whether $\sum \limits_{n=0}^\infty\frac{(-1)^{n+1}}{5n+1}$ is convergent, absolute convergent or divergent

Everything in red is edited To show, that the series is convergent we show at first, that $\color{red}{\lim \limits_{n \to \infty} \left(\dfrac{1}{5n+1}\right)}=0$. $\color{red}{\lim \limits_{n \to \...
1
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4answers
56 views

Converge Test on the series $\sum \limits_{n=0}^{\infty} \left(\frac{2n+n^3}{3-4n}\right)^n$

I want to show, that $a:=\sum \limits_{n=0}^{\infty} \left(\dfrac{2n+n^3}{3-4n}\right)^n$ is not converging, because $\lim \limits_{n \to \infty}(a)\neq 0 \; (*)$. Therefore, the series can't be ...
0
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1answer
57 views

Convergence for the serie $\sum_{n=2}^{\infty}(-1)^n \ln (1-\frac{1}{n^{\alpha}})$ for $\alpha \in \mathbb R$

In order to study the convergence of the serie of general term $u_n=(-1)^n \ln (1-\frac{1}{n^{\alpha}})$, I remark that for $\alpha \leq 0$, the sequence $u_n$ does not tend towards zero. Suppose ...
-1
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1answer
24 views

How to determine whether the following two infinite series converge absolutely, converge conditionally, or diverge. [closed]

I need some guidance on how to solve these, I'm not understanding series and sequences too well and I need an explanation that hasn't come from my lecturer. $$\sum_{k=1}^\infty \frac{\log k}{k^2}$$ $...
1
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0answers
21 views

Limit of Coefficient Sequence and Radius of Convergence

Show that if the power series $\sum_{n=0}^\infty a_n x^n$ has radius of convergence $R$ and if $\lim_{n \to \infty} |a_{n+1}/a_n|$ exists, then the value of this limit is $R$ I think there might be a ...
1
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2answers
87 views

Showing a series converges absolutely almost everywhere

Let $f:\mathbb{R}^m\rightarrow\bar{\mathbb{R}}$ be a Lebesgue integrable function with $\int |f|>0$. Show that the infinite series $\sum_n\frac{f(n\vec{x})}{n^p}$ converges absolutely almost ...
2
votes
2answers
78 views

Find the radius of convergene R for power series

For power series, find the radius of convergence R and determine if it is conditionally convergent, absolutely convergent, or divergent for $z = R$ and $z = −R$. $\sum_{i=0}^{\infty} e^n z^n$ I'm ...
0
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3answers
37 views

Existence of additional condition for Convergence of $\sum a_nb_n$ with monotonicity condition is dropped?

I know there are 2 theorem for convergence of $\sum a_nb_n$ which has following assumption. 1) If $\sum a_n$ is convergent and $b_n$ is monotonic and bounded. then $\sum a_nb_n$ is convergent. 2) If ...
1
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1answer
32 views

Are absolutely convergent sums exactly those which can be reordered? [duplicate]

If an infinite sum is absolutely convergent, its limit remains the same however the terms are permuted. Does a sequence of real numbers $(a_n)_{n\in\mathbb{N}}$ exist such that $\sum_n a_{\small P(n)}$...
0
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0answers
30 views

Poles of alternating series

Consider the function $f(\cdot)$ defined as follows, $$ f(x) = \sum_{k=0}^{\infty} a_k \left(\frac{-1}{x}\right)^k$$ where $a_0 = 1$ and $a_k > 0$ for all $i$. Assume the series converges in ...
1
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1answer
40 views

Confusion if the series converges or not (alternating series test)

I have to test for convergence and absolute convergence for the following series: $$\sum_{k=1}^{\infty} (-1)^k \frac{k}{1+2k^2}$$ Because of the alternating series test, I have to verify if the ...
3
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2answers
54 views

Find absolute convergence of $\sum_{n=1}^{\infty}\left[{\frac{\sin{\frac 1 n}+\cos{\left({n\pi}\right)}}{n}}\right]$

I have the following series: $$\sum_{n=1}^{\infty}\left[{\frac{\sin{\frac 1 n}+\cos{\left({n\pi}\right)}}{n}}\right]$$ My textbook asks to determine the simple and absolute convergence of the series....
0
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1answer
22 views

Interval of convergence, pointwise and absolute

Give the series $$\sum_{n=0}^{\infty} \dfrac{(x + 10)^n}{3^n (n+1)},$$ find the intervals which result in point-wise and absolute convergence. Applying the root test we have, $$L(x) = \lim\limits_{n ...
0
votes
1answer
219 views

Normed Space $X$ complete iff any absolutely comvergent series in $X$ converges

I'm studying functional analysis. I have trouble with the following proposition and its proof. Wonder if someone could help me with the following questions: Proposition: A normed space $X$ is ...
2
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0answers
86 views

Determine Convergence of $\sum_{n=1}^\infty \frac{(-1)^nn!}{(n+100)!}$

I know that $\frac{n!}{(n+1)!}$ can be reduced to $\frac{1}{n+1}$, but i'm not sure about this one. $$\sum_{n=1}^\infty \frac{(-1)^nn!}{(n+100)!}$$ In my notes, my professor reduced it to a p-...
0
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2answers
54 views

Why we ONLY use ratio test and not conditional convergence to determine the interval of convergence of an alternating series?

For example, consider $$S_n=\sum_{n=1}^{\infty} \frac{(-1)^n x^n} {\sqrt{n}}$$ While determining the interval of convergence, we use the ratio test to determine the interval in which the series ...
2
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1answer
82 views

Convergence of an infinite determinant

I'm stuck upon the following exercise from A Course of Modern Analysis by Whittaker: Show that the necessary and sufficient condition for the absolute convergence of the infinite determinant $$\...
2
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1answer
75 views

find when $\sum^{\infty}_{n=0} x^n \tan \left(\frac {x}{2^n}\right)$ is convergent

For which real numbers x is the series $$\sum^{\infty}_{n=0} x^n \tan \left(\frac {x}{2^n}\right)$$ convergent and how (i.e. absolutely/conditionally)? I have proved that the series converges ...
0
votes
1answer
77 views

Convergence of complex integrals: Necessary and Sufficient conditions.

Currently I am examining functions defined in the following way: $F(z)=\int f(z,t) dt\ $where the integral is along some curve $\gamma\\$ not necessarily closed. I want to know necessary and ...
1
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1answer
57 views

Examples of complete families of functions forming an absolutely convergent series

I am searching for some examples of complete families of functions $\left\{ \phi_m(t) \right\}_{m = 1}^\infty$ on $t \in [0, T]$ that form an absolutely convergent series: $$ \sum_{m = 1}^\infty |\...
1
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1answer
71 views

When the series of orthogonal functions converges absolutely?

It is given a family of functions $\phi_n(t)$ orthogonal in $[0, T]$, $0 < T < \infty$. What conditions must $\phi_n$ satisfy in order to have $$ \sum_{n = 1}^\infty |\phi_n(t)| \leq c < \...
1
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3answers
225 views

Does this series converge absolutely $\sum_{n=1}^{\infty}\frac{b^{n}_{n}\cos(n\pi)}{n}$

Let $\{b_n\}$ be a sequence of positive numbers that converges to $\frac{1}{2}.$ Determine whether the given series is absolutely convergent. $$\sum_{n=1}^{\infty}\frac{b^{n}_{n}\cos(n\pi)}{n}$$ ...
0
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1answer
34 views

Determine whether the given series is absolutely convergent or conditionally convergent

Consider the series $$\sum_{n=1}^\infty \log\left(1+\frac{1}{|\sin(n)|}\right).$$ Determine whether it converges absolutely or conditionally. I am trying to apply Cauchy condensation test, but I ...
-2
votes
1answer
67 views

Is this series conditionally convergent or absolutely convergent? $\sum_{k=1}^{\infty}\left(-1\right)^{k+1}\sin\left(\frac{1}{k}\right)$

This series is not absolutely convergent because \begin{align*} \lim_{k\rightarrow+\infty}\frac{\bigl|\left(-1\right)^{k+1}\sin\left(\frac{1}{k}\right)\bigr|}{\frac{1}{k}} & =\lim_{k\rightarrow+\...
0
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2answers
57 views

Conway's Complex Analysis: Radius of Convergence

1.3 Theorem. For a given power series $\sum_{n=0}^\infty a_n(z-a)^n$ define the number $R$, $0 \le R \le \infty$, by $$\frac{1}{R} = \limsup |a_n|^{1/n}.$$ Then...(b) if $|z-a| > R$, the terms of ...
6
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0answers
311 views

Series whose Cauchy product is absolutely convergent - A general example

Is there series that is divergent or conditionally convergent with absolutely convergent Cauchy product? Seems like there is a group of these examples! Perhaps finding divergent series with ...
0
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1answer
52 views

Assessing Whether the Series $\sum_\limits{n = 0}^\infty (-1)^n x^{2n} = 1 - x^2 + x^4 - x^6 + \dots$ Converges (conditional/absolute) or Diverges?

It's been a while since I've used the various tests for convergence (conditional/absolute) and divergence, and I can't remember which test needs to be used and how to assess whether the series $\sum_\...
1
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0answers
59 views

Convergence of sum on lattice in $\mathbb{C}$

I am trying to understand why the following sum converges $$\sum_{\lambda \in \Lambda\backslash\{0\}}\frac{1}{|\lambda|^3},$$ where $\Lambda=\{m+n\tau \mid m,n\in \mathbb{Z} \}$ with $\tau \in \...
1
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1answer
38 views

Absolute and conditional convergence of series with parameter

I have the following series: $$\sum_{n=2}^{\infty}\frac{(-1)^n}{(n+(-1)^n)^p}$$ I need to check for absolute and conditional convergence values, depending on the parameter $p$. Any tips on how to ...
1
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4answers
137 views

Series and Sequences - Absolute Convergence

I'm practicing for my final exams this week but the past year papers have no answers so I'm not sure if my answers are acceptable, was hoping someone would look at my proof and let me know if it is a ...
0
votes
1answer
61 views

Product with itself is not absolutely convergent $\sum_{1}^{\infty}\frac{(-1)^n}{\sqrt{n}}$

Show that the product of the series with itself is not absolutely convergent. $$\sum_{1}^{\infty}\frac{(-1)^n}{\sqrt{n}}$$ I tried looking at Cauchy product series. $$c_n= a_1b_n+...+a_nb_1\implies ...
1
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3answers
81 views

Absolute convergence of the series $\sum_1^{\infty} {(-1)}^n(\sqrt{n+1} -\sqrt{n})$

$\sum_1^{\infty} {(-1)}^n(\sqrt{n+1} -\sqrt{n})$ I want to know if this series diverge or if it is absolutely convergent or conditionally convergent. I used Leibniz' Criterion of alternating series ...
1
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2answers
269 views

What conclusion can we make if $f$ is a holomorphic function?

Suppose $f$ is holomorphic in an open neighbourhood of $z_0 \in \Bbb C$. Given that the series $$\sum\limits_{n=1}^{\infty} f^{(n)} (z_0)$$ converges absolutely, we can conclude that $(1)$$\ \ \ \...
0
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2answers
29 views

Divergence of positive series implies divergence of alternating series?

Consider the alternating series $\Sigma_{n=0}^{\infty} \left(\frac{2n+2}{2n+1}\right)^n (-1)^n $. The question is to assess its absolute convergence. If I set $u_n = \left(\frac{2n+2}{2n+1}\right)^n =...
1
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3answers
108 views

Does $\sum _{n=1}^{\infty }\left(-1\right)^{n+1}\left(1-cos\left(\frac{1}{\sqrt{n}}\right)\right)$ converges conditionally?

I'm trying to understand whether the following series converges absolutely, conditionally or diverges. $$\sum _{n=1}^{\infty }\left(-1\right)^{n+1}\left(1-cos\left(\frac{1}{\sqrt{n}}\right)\right)$$ ...
3
votes
0answers
56 views

Two identities involving Gregory coefficients and different arithmetic functions from an integral representation

This morning I've deduced two identities that involve Gregory coefficients $G_n$ invoking the so-called Schröder's integral formula (this is the Wikipedia's article for Gregory coefficients). The ...
1
vote
5answers
69 views

Testing $\sum\limits_{k=1}^∞(\frac{k+1}k)^{k^2}3^{-k}$ for convergence and absolute convergence

Test $$\sum_{k=1}^{\infty}\left(\frac{k+1}{k}\right)^{k^2}3^{-k}$$ for convergence and absolute convergence. We apply the ratio test for $\displaystyle \sum_{k=1}^{\infty}\left|\left(\frac{k+1}{k}\...
1
vote
2answers
60 views

If $\sum\limits_{n=1}^∞u_n^2$ is convergent, then $\sum\limits_{n=1}^∞\frac{u_n}n$ is absolutely convergent [duplicate]

If $\{u_n\}$ is a sequence of real numbers and the series $\displaystyle\sum_{n=1}^{\infty}u_n^2$ is convergent, prove that the series $\displaystyle\sum_{n=1}^{\infty}\frac{u_n}{n}$ is absolutely ...
0
votes
1answer
50 views

Finding a sequence $a_n$ that diverges such that $\|a_n\|$ converges (in $\mathbb{R}$)

Finding a sequence $a_n$ that diverges such that $\|a_n\|$ converges (in $\mathbb{R}$) I am having a hard time finding an example that works. An example or hint would be greatly appreciated.
1
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0answers
22 views

Can a series be non absolutely convergent at point beyond its radius of absolute convergence? [duplicate]

Let $\sum a_nx^n$ be a series, where $a_n$'s are not necessarily positive & let $R>0$ be its radius of absolute convergence. Then for $x=r>R$, $\sum |a_n|r^n$ will be divergent, but my ...
1
vote
2answers
43 views

Help with the AST

Can someone help me verify if this is true? When using the Alternating Series Test (AST), do I need to look at the absolute values of the terms and see if they converge to confirm that the series is ...