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Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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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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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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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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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 ...
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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 ...
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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 ...
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 ...
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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 ...
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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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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 ...
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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 ...
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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: $$A(x) = 1 + \frac{x^3}{2\times3}+\frac{x^6}{2\times3\times5\times6} + \frac{x^9}{2\times3\times5\times6\times8\times9} +\space ...$$...
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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 ...
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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 ...
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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 ...
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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 ...
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 $0$ ...
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 ...