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Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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3answers
258 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. $$\sum_{n=1}^{\infty}\frac{b^{n}_{n}\cos(n\pi)}{n}$$ ...
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1answer
35 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
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1answer
69 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
60 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
326 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
53 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
73 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
41 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
192 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
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1answer
62 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
86 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
318 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)$$\ \ \ \...
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2answers
30 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
117 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
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0answers
63 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
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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
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2answers
63 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
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1answer
51 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.
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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
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2answers
53 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 ...
1
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1answer
42 views

Find the domain of $A(x) = 1 + \frac{x^3}{2\times3}+\frac{x^6}{2\times3\times5\times6} + \frac{x^9}{2\times3\times5\times6\times8\times9} +\space …$

I'm having trouble in finding the domain of the following series: $$A(x) = 1 + \frac{x^3}{2\times3}+\frac{x^6}{2\times3\times5\times6} + \frac{x^9}{2\times3\times5\times6\times8\times9} +\space ...$$...
0
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1answer
63 views

Dominated convergence theorem random variable definition

For nonnegative RV X s.t. E|X|<$\infty$, $X_n=t_n I_{X>t_n}$ for a sequence of $t_n \to \infty$, since $X_n \to 0$ , then $E(X_n)=t_nP(X>t_n)$ converge to zero as well. This is from the ...
0
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3answers
74 views

Absolutely convergent series of complex numbers

If $\sum_{n=0}^{\infty} a_n$ is an absolutely convergent series of complex numbers and $a_n \ne -1$ $\forall$ $n$, prove that the series $\sum_{n=0}^{\infty} \frac{a_n}{1+a_n}$ is absolutely ...
0
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1answer
20 views

Clarification of an example on absolute convergence test of a series

When testing whether or not series converge absolutely with a convergent $p$-series, we have to test it against $\sum_{n=0}^{\infty}|a_n|$. In one of the text book examples, I'm having trouble ...
0
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1answer
52 views

Show that the series $\sum_{n=0}^\infty (-1)^n \frac{1}{(n+1)^p}$ converges and converges absolutely

Show that the series $\sum_{n=0}^\infty (-1)^n \frac{1}{(n+1)^p}$ converges for $p > 0$ and converges absolutely for $p > 1$. I'm a little confused on this. The fact that it's an alternating ...
0
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2answers
109 views

Proof of a series $\sum_{n=0}^{\infty} x_n$ is absolutely convergent. [duplicate]

Let $\{x_n\}$ be a given series such that it satisfies the following conditions for all sequence $\{y_n\}$ in real numbers converging to $0$. It is given that the sequence $\{y_n\}$ converges to $0$ ...
4
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2answers
59 views

Absolute convergence of $\sum_{n=1}^{\infty}z_n^2$

There are two complex series given ($z \in \mathbb{C}$. We know that: $$\sum_{n=1}^{\infty}z_n^2$$ converges absolutely. We are to show that $$\sum_{n=1}^{\infty} \frac{z_n}{n}$$ also converges ...
1
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1answer
40 views

Convergence and absolute convergence of sums

Examine if the series converge $\sum_{n=1}^\infty \sin^2(\frac{1}{n})$. For which $a$$\in \mathbb{R^+}$ the series converge $$\sum_{n=1}^\infty\frac{\sqrt{n^4+1}}{n^a}?$$ Also for which $a$ the ...
0
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1answer
39 views

Proving a space is not complete by finding an absolutely convergent series

For $ v ∈ l^2(\mathbb{N}, F) $ define $$ ||v||_w=\sum_{k=1}^\infty |v_{[k]}|/2^k$$ as a norm, Is $ l^2(\mathbb{N}, F) $ with the norm $||v||_w $ a complete space? I am trying to find a series of ...
0
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0answers
18 views

Prove absolute convergence given the following inequality

We are given the following inequality: $|e^{x}-1|\leq|x|e^{|x|}$ where $x \in \mathbb C$ Prove that if $\sum_{n\in \mathbb N} a_{n} $ converges absolutely in $\mathbb C$ then: $\sum_{n\in \mathbb N}...
2
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5answers
451 views

Does it make mathematical sense to do an absolute convergence test if the original series diverges?

Reason I ask I know a series can converge but then when you apply the absolute convergence test it may diverge. I understand this part. One concludes absolute convergence is a stronger condition! ...
1
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1answer
117 views

Conditional convergent improper Riemann integral vs. Lebesgue Integral

For all this, I'm thinking on functions defined on $\mathbb{R}$. I've already read that if a function $f$ is absolute improper Riemann integrable, then $f$ is Lebesgue integrable and both integral ...
0
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4answers
66 views

Prove $\int_{\pi}^{\infty}\frac{\cos(x)}{x}dx$ is convergent

I am supposed to prove that the integral in the question is convergent, but I seem to be stuck on finding an upper bound. It's obvious that the integrand is not positive for all $x \in [\pi, \infty[$ ...
1
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1answer
92 views

Every rearrangement of an absolutely convergent series converges to the same sum (Rudin)

What guarantees $\sum_{i=n}^m |a_i|$ will not be less than $|s_n-s_n'|$; hence $|s_n-s_n'| \ge \epsilon$? May someone explain, please?
1
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1answer
28 views

Does one-sided derivative of real power series at edge of domain of convergence

Supoose that the power series $f(x)=\displaystyle\sum_{i=0}^\infty a_ix^i$ has radius of convergence $R=1$, and that $f$ converges at $x=1$. I am aware that $f'$ may not exist at $1$, but must the ...
1
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2answers
94 views

Convergence and absolute convergence of a series in ℝ

I have been given the following definition: $\rule{17cm}{0.4pt}$ Let $\{a_n\}$ be a sequence in $\mathbb{R}$. The series: $$\sum_{n=0}^\infty a_n$$ is $\textbf{convergent}$ if the sequence $\{s_m\}$ ...
4
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6answers
141 views

Convergence/absolute convergence of $\sum_{n=1}^\infty \left(\sin \frac{1}{2n} - \sin \frac{1}{2n+1}\right)$

Does the following sum converge? Does it converge absolutely? $$\sum_{n=1}^\infty \left(\sin \frac{1}{2n} - \sin \frac{1}{2n+1}\right)$$ I promise this is the last one for today: Using Simpson'...
1
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2answers
215 views

Convergence/absolute convergence of the series $\sum_{n=2}^{\infty} (-1)^{n+1}\frac{\ln (n)}{n}$

Does the series $\sum_{n=2}^{\infty} (-1)^{n+1}\frac{\ln (n)}{n}$ converge? If so, does it converge absolutely? My attempt: Because $$\left(\frac{\ln x}{x}\right)' = \frac{1-\ln(x)}{x²} > 0$$ ...
0
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2answers
61 views

proving that a sum is absolutely convergent iff another sum is absolutely convergent

I'm having trouble with the following question: Prove that $\sum_{n=1}^\infty\left(\frac{1}{a_{n+1}}-\frac{1}{a_n}\right)$ is absolutely converges if and only if $\sum_{n=1}^\infty(a_{n+1}-a_n)$ ...
1
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3answers
68 views

Does $\sum\limits_{k=2}^\infty\frac{(-1)^{2^k}x^{2k}}{(2k+3)!}$ converge absolutely?

Let $\sum\limits_{k=2}^\infty\frac{(-1)^{2^k}x^{2k}}{(2k+3)!}$, $x\in\mathbb{R}$. Does this series converge absolutely? $$\sum\limits_{k=2}^\infty\frac{(-1)^{2^k}x^{2k}}{(2k+3)!}$$ converges ...
2
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2answers
303 views

Absolute convergence of $\sum\limits_n\left( \log (1-\frac{z}{n})^{n^{k}} + \sum\limits_{m=1}^{k+1}\log e^{n^{k-m}z^m/m} \right)$?

I'm having trouble verifying my appoarch to the problem in $(1)$, much of efforts can be seen in the sections titled $\text{Lemma}$, I'm specifically stuck where $(1.6)$ would the absolute convergence ...
4
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2answers
177 views

Prove that the series $\sum\limits_{n=-\infty}^{+\infty}f(x+n)$ converges absolutely for a.e. $x \in \mathbb{R}.$

Problem: Let $f$ be a Lebesgue integrable function on $\mathbb{R}.$ Prove that the series $$\sum\limits_{n=-\infty}^{+\infty}f(x+n)$$ converges absolutely for a.e. $x \in \mathbb{R}.$ What ...
0
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2answers
101 views

Absolute convergence of series $\sum\limits_{n=2}^\infty{(-1)^{n-1}\over (n+(-1)^n\sqrt{n})^{2\over 3}}$

Exam absolute convergence of series $$\sum\limits_{n=2}^\infty{(-1)^{n-1}\over (n+(-1)^n\sqrt{n})^{2\over 3}}$$ WolframAlpha says it diverges, but I don't know how to show it. As it's not monotonic $\...
0
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1answer
35 views

If $\sum (a_j)^2$ abs converge , does $\sum a_j / (j+1) $ that too?

Suppose that $\sum (a_j)^2$ converge absolute, how can I prove that $\sum a_j / (j+1)$ does that too? I don't exactly know where to start to prove this. Can somebody help me or give a hint?
2
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0answers
28 views

System stability and $z$-transform

I have been struggling to understand the relation between $z$-transform and the study of analysis but there is something that puzzles me. Stability of a discrete time system is happens if and only if ...
0
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2answers
115 views

Show that the Taylor series for the principal part of $\log(1+z)$ converges absolutely for $|z|\le1$

Show that the Taylor series for the principal part of $\log(1+z)$ converges absolutely for $|z|\le1$ the Taylor series for $$\log(1+z)=z-\frac{z^2}{2}+\frac{z^3}{3}-\frac{z^4}{4}+......\sum _{n=1}^\...
1
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4answers
49 views

Checking for convergence of $\sum_{n=0}^\infty\frac{(-1)^nn}{n^2+1}$

I have the following series $\sum_{n=0}^\infty\frac{(-1)^nn}{n^2+1}$ which I am checking for convergence. My working: $\lim_{n\to\infty}|a_n|=0$ where $a_n=\frac{(-1)^nn}{n^2+1}$. Doesn't that ...
1
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0answers
32 views

Bounding support of a probability measure by calculating radius of convergence of Stieltjes transform given by a sum

The general idea It is a well-known fact that the Stieltjes transform $s_\mu$ of a probability measure $\mu$ on $\mathbb R$ is analytic on $\mathbb C\setminus\text{supp}(\mu)$. So if I wanted to ...
0
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3answers
128 views

Need a second help in understanding a step in matrix representation of bounded linear operators.

In completion to this question: Need A help in understanding a step in matrix representation of bounded linear operators. The book said: "Now, $$A \phi_{j} = \sum_{k}<A \phi_{j},\phi_{k}> \...
0
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2answers
44 views

Find $z_n$ divergent, so that $|z_n|$ converges?

Do you know a divergent series or sequence $(z_n)_{n\in\mathbb{N}}$ ($z_n\in\mathbb{C}$), which absolut value $(|z_n|)_{n\in\mathbb{N}}$ converges? I was not able to find one... only in the other ...