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Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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15 views

absolut and uniform convergence of complex power series on an open disk imply absolut and local uniform convergence on the whole disk of convergence [duplicate]

I'm stuck in the proof of the following theorem of complex analysis: There is a $R\in\mathbb{R}_{0}^{+}\cup\{\infty\}$ s.t.: (i) $\sum_{k=0}^{\infty} a_k (z-z_0)^k$ converges absolutely and ...
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0answers
21 views

On the convergence of a two-sided series

Background Let $\left\{ {{a_n}} \right\}_{ - \infty }^\infty $ be a two sided sequence (is there a more proper term?) of complex numbers. As far as I know (please correct me if I am wrong) we say ...
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2answers
31 views

Find radius of convergence of the power series.

Find the radius of convergence of power series $$ \sum_{n=0}^{\infty} 2^{2n} x^{n^2}$$ A)1 B)2 C) 4 D)1/4 I try to apply ratio and root test ( Cauchy–Hadamard theorem ) .but they ...
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2answers
42 views

Let $X$ be a Banach space then every absolutely convergent series in $X$ converges in $X$

my trial Let $\sum x_k$ be absolutely convergent in $X$ $\implies$ $\sum \|x_k \|$ converges in $\mathbb{R}$ $\implies$ $\forall \epsilon >0, \exists N(\epsilon)$ st $\forall n>N(\epsilon)$. ...
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2answers
49 views

Hint requested for: If $\sum_{n=0}^{\infty} a_n x^n$ converges for some $x_0$, then it converges uniformly and absolutely on $[-a, a]$ with $a<|x_0|$?

I would like to prove If $\displaystyle\sum_{n=0}^{\infty} a_n x^n$ converges for some $x_0$, then it converges uniformly and absolutely on $[-a, a]$ with $0<a<|x_0|$. (Sorry not enough ...
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1answer
42 views

Difficulty regarding understanding a proof of multiplication of infinite series

I am referring to article no $3.50$, from Principles of Mathematical Analysis by Walter Rudin. The theorem is Let $\sum_{n=0}^\infty a_n=A$ and $\sum_{n=0}^\infty b_n=B$ be two convergent series. ...
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1answer
29 views

Convergence of sum using ratio test

Determine the values of x ∈ R for which the following series converge: $\sum_1^\infty \frac{x^nn^n}{n!}$ My attempt: I used the ratio test to obtain $|x|< \frac{1}{e}$ as the interval where it ...
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1answer
31 views

Dirichlet test for complex sequences

The sum $$\sum_{k=1}^\infty a_k b_k$$ converges when $a_k$ is monotonically decreasing and $$B_n=\sum_{k=1}^n b_k$$ is finite/bounded $\forall n$. This follows from summation by parts. I'm now ...
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1answer
317 views

Has the Riemann Rearrangement Theorem ever helped in computation rather than just being a warning?

A decent course in elementary analysis will eventually discuss series, absolute convergence, conditional convergence, and the Riemann Rearrangement Theorem. However, in any presentation I've seen, in ...
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3answers
46 views

Show with the direct Comparison Test that $\sqrt[n]{|a_n|}\leq\theta$ converges absolutely

Let $\sum_{n=1}^{n=\infty}{a_n}$ be an infinite series of real numbers. There is a $\theta$ with $0<\theta<1$ and a $n_0 > 0$, so $$\sqrt[n]{|a_n|}\leq\theta$$ for all $n \ge n_0$. Show that ...
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1answer
30 views

Absolute or conditional convergence?

Determine whether the series: $$\sum_{n=1}^\infty (-1)^n \frac {2n^2+3n+4} {2n^4 + 3}$$ converges absolutely, conditionally or diverges. I know the series converges conditionally using alternating ...
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5answers
119 views

Convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n\sqrt[n]n}{\log\ \ n}$.

I am analizing the convergence, absolute convergence and conditional convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n\sqrt[n]n}{\log\ \ n}$. I proved already that the series $\sum_{n=1}^{\...
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2answers
44 views

Infinite Series Diverges By Divergence Test But Converges By Limit Comparison Test

Image of My Work I understand why this infinite series diverges by the divergence test but I can't find fault in my limit comparison test which says it diverges. Please help. Thanks P.S. if my ...
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0answers
43 views

Does convergence of $\Gamma(x)$ imply convergence of $\Gamma(z)$? Is it generalisable?

If we have the gamma function in integral form $\Gamma(x)=\int_\limits0^\infty e^{-t}t^{x-1}dt$ and have proven that it converges for real $x>0$ (to a point), can we then immediately conclude that $...
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0answers
25 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" ...
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2answers
32 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
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2answers
295 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|\...
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1answer
21 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, ...
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1answer
29 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)...
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2answers
57 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 ...
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1answer
24 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
27 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
39 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 ...
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1answer
63 views

(Proof Verification) Examining whether the series $\sum \limits_{n=0}^\infty \dfrac{(-1)^{n+1}}{5n+1}$ 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 \...
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4answers
50 views

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

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 ...
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1answer
29 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 ...
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1answer
21 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. $$\sum_{k=1}^\infty \frac{\log k}{k^2}$$ $...
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0answers
20 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 ...
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2answers
55 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 ...
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2answers
62 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 ...
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3answers
33 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 ...
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0answers
28 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 ...
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1answer
34 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 ...
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2answers
53 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....
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1answer
20 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 ...
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1answer
102 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 ...
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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-...
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2answers
26 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 ...
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1answer
49 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
70 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 ...
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1answer
40 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 ...
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1answer
53 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 |\...
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1answer
50 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 < \...
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1answer
32 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 ...
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1answer
61 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+\...
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2answers
53 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
255 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
48 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
35 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
31 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 ...