Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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64 views

Michael Spivak “Calculus 3rd Edition” Chapter 23. Infinite Series Theorem 9 about $\sum_{n=1}^{\infty} a_n \cdot \sum_{n=1}^{\infty} b_n$.

I am reading "Calculus 3rd Edition" by Michael Spivak. What does $\sum_{i \text{ or } j > L} |a_i| \cdot |b_j|$ mean precisely? What is the definition of $\sum_{i \text{ or } j > L} |a_i| \...
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2answers
56 views

Prove that $\sum |x_k| < \infty \iff \sum |x_k|^p < \infty$

I'm trying to solve this problem about series. Could you please verify if my attempt is fine or it contains logical gaps/mistakes? I'm grateful for your help! Let $\mathbb K$ denote $\mathbb R$ or $...
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1answer
29 views

Convergence of integral: $ \int_{0}^{\infty} \frac{x^{p}\sin{x}}{1+x^{q}}\ dx $

I have problem with determining about convergence of integral below.I firstly tried to determine about absolute convergence by limit test but i cannot find relation between parameter $p$ and $q$.For ...
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1answer
24 views

Convergence of integral: $ \int_{0}^{\infty} \frac{\sin{\left(x+\frac{1}{x}\right)}}{x^{a}}\ dx $

I need to determine about convergence of integral below.I found out that it doesn't converge absolutely but i have problem with determining about conditional convergence.I am not sure how to use ...
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2answers
34 views

Convergence of: $\sum _{n=1}^{\infty }{\left(-1\right)^{n}\frac{2^{n}\sin^{2n}{\left(x\right)}}{n}}$

I solved that it converges absolutely on $\bigcup \left(-\frac{\pi}{4}+k\pi,\frac{\pi}{4}+k\pi\right) \text{for all k belongs to integer numbers}$ I don't know how to deal with "non-absolute" ...
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3answers
67 views

Show that $2\cos z = e^{iz} + e^{-iz}$. Do I need absolute convergence?

I'm reading Conway's "Functions of one complex variable". Define $$e^z =\sum_{n=0}^\infty \frac{z^n}{n!}$$ $$\cos z =\sum_{n=0}^\infty \frac{(-1)^nz^{2n}}{(2n)!}$$ It is already shown that these ...
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2answers
45 views

Is my understanding of the proof of rearrangement theorem correct?

I'm reading the proof of rearrangement theorem. Could you please verify if my understanding of the last part of the proof is correct or not? The authors said The inequality $(8.3)$ implies the ...
1
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1answer
57 views

Let $\alpha >0$ and $ \varepsilon > 0 $. Then $\sum_{n = 0}^{ + \infty} \frac{1}{(1 + |n^2 + n - \alpha |) (1 + n + n^2)^{\varepsilon}} < + \infty $?

Let $\alpha >0$ and $\varepsilon > 0$. Then $\sum_{n = 0}^{ + \infty} \frac{1}{(1 + |n^2 + n - \alpha |) (1 + n + n^2)^{\varepsilon}} < C < + \infty $ ? where $C $ does not depends on $\...
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1answer
29 views

An absolutely summable series but not 1-summable? [closed]

I am trying to find an example of an absolutely summable series ($\Sigma|{a_j}|<\infty$) but not 1-summable series ($\Sigma j|{a_j}|<\infty$). I know that all 1-summable series are absolutely-...
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2answers
37 views

Exercises to see if certain series converge.

Does the series $\sum \frac{2^k+1}{3^k}$ converge? Does the series $\sum (-1)^k\frac{1}{k^2}$ converge? Does the series $\sum \frac{2^k+2^{2k}}{4^k}$ converge? Does the series $\sum \frac{k!}{(...
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3answers
50 views

why converges absolutely needed for that prove *?*

Theorem: If $\sum_{k=1}^{\infty} a_k$ converges absolutely, then any rearrangement of this series converges to the same limit. I will try to prove without using converges absolutely condition. ...
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2answers
38 views

Radius of convergence of power series and absolute convergence

We know that if the power series $\sum_{k=0}^\infty a_kz^k, a_k, z \in \mathbb{C}$ has a radius of convergence $R$ then it converges absolutely for $|z| < R$ and diverges for $|z| > R$ but the ...
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1answer
36 views

Convergence and Divergence in $2^{-\ln n}$

The series that I need to check is $2^{-\ln n}$ and $3^{-\ln n}$. For the second case, after doing the preliminary test (which gave me zero so that I may proceed further), I tried to solve it by ...
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1answer
30 views

Show that the next double power series is convergent

Show that $$\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}[k|\beta +1]_n\frac{|t|^k}{k!}\frac{|x|^n}{n!}$$ is convergent $\forall t \in \mathbb{R}$, and $x, \beta \in \mathbb{R}$ such as $|\beta x|<1$, ...
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2answers
105 views

Convergence of $\sum_{k=1}^{\infty}\frac{\cos(\theta k)}{\sqrt{k}}$

Say if the following series $$ \sum_{k=1}^{\infty} \frac{\cos(\theta k)}{\sqrt{k}} $$ for $θ \in \mathbb{R}$ is convergent. Is it absolutely convergent? I don't know how to approach this problem. ...
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1answer
55 views

Sum of the Riemann-Zeta function at the integers

If you consider the series of reciprocal positive integers (the harmonic series), the sum diverges. However, some subsets of this series, such as the reciprocal cubes, will sum to something finite. I ...
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1answer
31 views

Absolute convergence of $\sum_{k=0}^{+\infty}a_{n}(x+2)^n$

Define the series $$\sum_{n=0}^{+\infty}a_{n}(x+2)^n.$$ All we know is that it converges in x=4. Why can't I say it converges absolutely in x=4?
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2answers
60 views

If $\sum a_n$ converges absolutely, then so does $\sum f(a_n).$

Hi I am stuck on this problem. (please dont give the solution I just need some help to formalism my solution) Let f: R to R be differentiable, with continuous derivatives and f(0) = 0 If $\sum{a_{n}}$...
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1answer
23 views

series convergence imply nx approaches 0

Proof: For a decreasing sequence of positive reals, show that if the sum converges, then $nx_n \to 0$ but the converse is not true The first part I just assumed a positive limit the series converge ...
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1answer
47 views

Definitions of sin and cos using the exponent

I follows these steps: Define $e^z:=\sum_{k=0}^{\infty}\frac{z^k}{k!}$. Show thatthe series is absolutely convergent. Define $\sin(z):=\frac{e^{iz}-e^{-iz}}{2i}$, and $\cos(z):=\frac{e^{iz}+e^{-iz}}{...
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16 views

Absolute convergence of series variant of the geometric series

So I want to prove whether the following series converges absolutely or not: $$\frac{1}{2}\sum_{n=0}^\infty (n^2+3n+2)q^n$$ where $ q \in \mathbb{C}, \mid q\mid<1.$ My attempt was: $$\frac{1}{2}...
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33 views

Prove that the sum of a series is differentiable

Prove that the series $$\sum_{k=2}^∞ \sin (kx)/k\ln^2(k)$$ is absolutely and uniformly convergent on $\mathbb{R}$. If the sum of the series is denoted by $f(x)$ prove that $f$ is differentiable at ...
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3answers
48 views

Convergence of the series below

$$\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt{n}}$$ I did: $$\lim_{n\to \infty}\Biggr\vert\frac{(-1)^n}{\sqrt{n}}\Biggr\vert$$ $$\lim_{n\to \infty}\frac{1}{n^\frac{1}{2}}=0<1$$ So diverges by the Ratio ...
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1answer
25 views

Grouping the Summation

Let $a_n \in \mathbb{C}$ and consider $\sum a_n$ and grouping as $\sum (a_n + a_{n+1})$. Under what assumptions we can claim absolute convergence of grouped sum implies convergence of the original sum?...
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1answer
47 views

Is $\ell^1$ complete with this norm?

For $x \in \ell^1$ we set $\Vert x\Vert = \sup\limits_{N \in \mathbb{N}}|\sum\limits_{n=1}^{N}x_n|$. One can easily see that this is a norm on $\ell^1$. I was wondering if this space is now complete. ...
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3answers
53 views

Why would this series Absolutely Converge using Root Test?

So I was working on this problem and I got diverge, since my answer was greater than 1. The Limit was > 1, using the root test. $$\sum\limits_{n=4}^\infty (1 +\frac{1}{n})^{-n^2}$$ I ended up with ...
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1answer
31 views

Could I get an explanation on why this would conditionally converge?

$$\sum\limits_{n=2}^\infty \frac{\cos(n\pi)}{\ln(n)^2}$$ I'm not sure how this would conditionally converge, according to my calculations I would assume it's absolutely converge.
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1answer
61 views

$\sum u_v$ converges absolutely iff $\sum \log(1 + u_v)$ converges absolutely

I have trouble understanding the following proof of a fact in complex analysis. Assume $(u_v)_{v\geq1}$ is a sequence of complex numbers and $(u_v) \neq -1$ for all $v$. Then we have the following ...
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1answer
53 views

Switching Integral and Sum

I want to proof that I can switch this Sum and Integral $\sum\limits_{n=1}^\infty\int\limits_{0}^\infty t^{z-1} e^{-nt}dt~~$ for $~ 1 < Re(z) $ to sum it after over n. I tried to use the ...
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1answer
27 views

Shouldn't we check for conditionally convergent in ratio test done to see the intervals of convergence in power series? [closed]

(By A(n) I mean the power series)I understood that we use absolute value of A(n+1)/A(n) in ratio test because A(n) isn't neccessarily a positive value. We know when there is a limit of absolute value ...
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1answer
44 views

Can the Ratio & Root Tests show divergence directly?

An infinite series ⅀ $a_n$ is absolutely convergent if ⅀ $|a_n|$ is convergent. However, just because the absolute value of a series isn't convergent by some test doesn't mean it can't be ...
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3answers
32 views

Is $\sum_{n=1}^{\infty} \frac{(-1)^n(3n^3+4n^2)}{4+2n^5}$ divergent, conditionally convergent, or absolutely convergent?

The first thing I did was create a sequence ($A$) for what was inside of the sum, then I created another sequence ($B$) that is related to $A$. $$A=\frac{(-1)^n(3n^3+4n^2)}{4+2n^5}$$ $$B=\frac{1}{n^2}...
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2answers
210 views

Why are the sums of absolutely convergent series not affected by changing the order of summation?

Here is my attempt to understand this: Let $$ \sum_{n=1}^{\infty}\left(-1\right)^{\left(n+1\right)}a_n $$ be an alternating series now the infinite sum is defined as the limit of the sequence of ...
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2answers
62 views

Proof for radius of convergence

I have the following two problems related to ordinary differential equations (power series) a. Define the radius of convergence of a series $\sum_{n=0}^{\infty}{a_{n}(x-x_{0})}^n$ for general $(...
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1answer
69 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
67 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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22 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
51 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
65 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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49 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 ...
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1answer
36 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
40 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
45 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 ...
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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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45 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
61 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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34 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
67 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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18 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
28 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 ...