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Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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1answer
37 views

$\sum |a_n|<\infty$ and $|\sum b_n|<\infty$ implies $|\sum a_n b_n| <\infty$

Suppose $\sum_{n} a_n$ converges absolutely and $\sum_{n} b_n$ is any convergent series. Then $$\sum_n a_nb_n$$ is convergent. Proof: Since $\sum_{n} b_n$ is convergent, we can choose $N_0$ s.t $\...
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1answer
56 views

Does the series corresponding to a Cauchy sequence **always** converge absolutely?

Let $X$ be a normed vector space and consider a Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ in $X$. Is it true that the corresponding series of our Cauchy sequence, $\sum_{i=1}^\infty x_i$, always ...
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0answers
20 views

Divergence of $\sum_{n=0}^\infty |\sin \omega n|$

I'm looking for a simple argument to show that $$ \sum_{n=0}^\infty |\sin \omega n| $$ does not converge for $\omega \neq k \pi$, $k \in \mathbb{Z}$. If $\omega = \frac{1}{b}\pi$ with $b \in \...
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1answer
44 views

Why does this inequality hold? (“Complex Analysis” by Kunihiko Kodaira.)

I am reading "Complex Analysis" by Kunihiko Kodaira. I cannot understand why the following inequality holds. Please tell me the reason why this inequality holds.
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1answer
57 views

If every rearrangement of the series converges uniformly then the series converges absolutely uniformly

Let $I \subset \mathbb{R}$ and for $\forall n \in \mathbb{N}: f_n \in C(I, \mathbb{R})$. Prove that if for any $\sigma:\mathbb{N} \rightarrow \mathbb{N}$ bijection, the series $$\sum_n f_{\sigma(n)}$$ ...
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0answers
30 views

The proof of Taylor's theorem

Studying Analysis with 'Principles of Mathematical Analysis Third edition' written by Walter Rudin, I got some trouble from the proof of Taylor's theorem. The theorem and the proof from the book are ...
0
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1answer
32 views

Convergence when ratio test=1

When using the ratio test for absolute convergence of a series $\sum_{n=1}^\infty a_{n}$, if the limit of the ratio $$|a_{n+1}|/|a_{n}|=1$$ when $n \rightarrow \infty$, the fate of the series is ...
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2answers
34 views

Find the convergence radius for this power series

The Problem: Find the convergence radius of $\sum_{n=0}^{\infty} \frac{n}{5^{n-1}} z^{\frac{(n)(n+1)}{2}}$ My attempts to find a solution I apply either the ratio test and end up with this ...
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1answer
35 views

Help with convergence tests for series

I have a few questions to ask about series and convergence tests. I have been struggling to study everything fully and if someone can give me advices I will be really thankful.This is what I know so ...
0
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1answer
31 views

Prove that $\sum_{i \in S} c_i \leq \sum_{i \in S'} c_i$ if $S \subset S' \subset \mathbb{N}$.

Is the following proof correct or not? Please tell me better proof. Let $\{c_n\}$ be a sequence of positive numbers such that $\sum c_n$ converges. Let $S \subset S' \subset \mathbb{N}$. ...
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0answers
40 views

About $f(x) = \sum_{x_n < x} c_n$ in Remark 4.31 on p.97 in “Principles of Mathematical Analysis 3rd Edition” by Walter Rudin.

I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin. In Remark 4.31 on p.97, Rudin wrote this symbol $$\sum_{x_n < x} c_n.$$ What is the definition of this symbol ...
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3answers
56 views

Concluding whether $(y_n)_n$ is a bounded sequence

Suppose $(y_n)_n$ is a sequence in $\mathbb{C}$ with the following property: for each sequence $(x_n)_n$ in $\mathbb{C}$ for which the series $\sum_n x_n$ converges absolutely, also the series $\sum_n ...
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0answers
33 views

Convergence of $a_k=\sum_{k=5}^{+\infty}(-1)^k({3\over2})^{-k}(k^2+5)\sin(k+5)$

Firstly, $\lim_{k \to +\infty}a_k=0$,so necessary condition for convergence is satisfied.If we start to study absolute convergence we have : $$a_k=({3\over2})^{-k}(k^2+5)|\sin(k+5)|\leq({3\over2})^{-k}...
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1answer
62 views

If $a_n$ doesn't have any subsequence that converges, can $|a_n|$ converge?

If a sequence $a_n$, $n\in\mathbb{N}$ doesn't have any convergent subsequence can $|a_n|\rightarrow a$, $a\in[0,\infty)$? My intuition says that this isn't possible but I'm not sure how to prove it..
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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
23 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
36 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
50 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
51 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
52 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 ...
0
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1answer
32 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
326 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 ...
0
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1answer
36 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
124 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
48 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
44 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
27 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
34 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
303 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
23 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
36 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
62 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
25 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
30 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
41 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
64 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
52 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 ...
0
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1answer
38 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
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
69 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
69 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
34 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
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
37 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
votes
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....
0
votes
1answer
21 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
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1answer
171 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 ...