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Questions tagged [absolute-convergence]

This tag is for questions related to absolute convergence of a series.

This tag is for questions related to absolute convergence of a series. A series $\sum a_n$ is absolutely convergent if and only if $\sum |a_n|$ converges. Notice absolute convergence implies convergence but not vice versa; a series that is convergent but not absolutely convergent is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series.

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If every absolutely convergent series is convergent then $X$ is Banach

Show that A Normed Linear Space $X$ is a Banach Space iff every absolutely convergent series is convergent. My try: Let $X$ is a Banach Space .Let $\sum x_n$ be an absolutely convergent series .Consider $s_n=\sum_{i=1}^nx_i$. Now $\sum…
Learnmore
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Why do we ask for *absolute* convergence of a series to define the mean of a discrete random variable?

If $X$ is a discrete random variable that can take the values $x_1, x_2, \dots $ and with probability mass function $f_X$, then we define its mean by the number $$\sum x_i f_X(x_i) $$ (1) when the series above is absolutely convergent. That's the…
Amelian
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Does absolute convergence of a sum imply uniform convergence?

Suppose I have a series $\sum_{n = 0}^{\infty} f_{n}(x)$ which converges absolutely to a function $f(x)$. Does the series converge uniformly to $f(x)$? I want to say this follows from Dini's Theorem, but I can't seem to see how.
Shayla
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$b_n$ bounded, $\sum a_n$ converges absolutely, then $\sum a_nb_n$ also

a) Prove that if $\sum a_n$ converges absolutely and $b_n$ is a bounded sequence, then also $\sum a_nb_n$ converges absolutely. I wanted to use the comparison test to show it's true, but I think I got something mixed up. Here's what I have so…
Clash
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An absolutely convergent series of rational numbers which does not converge to a rational number

A standard theorem concerning series of real numbers states that every absolutely convergent series of real numbers converges. I would like to know a counterexample to this statement when we are dealing only with rational numbers. More precisely, I…
José Carlos Santos
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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 person or within a text, the discussion sort of…
user123641
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A series converges absolutely if and only if every subseries converges

Question: A subseries of the series $\sum _{n=1}^\infty a_n$ is defined to be a series of the form $\sum _{n=1}^\infty a_{n_k}$, for $n_k \subseteq \Bbb N$. Prove that $\sum _{n=1}^\infty a_n$ converges absolutely if and only if each subseries $\sum…
jreing
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Absolute convergence when all the rotated series converge

The question here might be standard in some textbook. Let $a_n, n\ge1$ be a series of real numbers. It is evident that if $\displaystyle \sum_{n\ge 1} |a_n|<+\infty$, then $\displaystyle \sum_{n\ge1} e^{2n\pi i t}a_n$ converges for all $0\le t<…
Pengfei
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Does $\ \sum_{n=1}^{\infty} \frac{\sin(2^n x)}{n}\ $ converge for all $x\ ?$

Does $\ \displaystyle\sum_{n=1}^{\infty} \frac{\sin(2^n x)}{n}\ $ converge for all $x\ ?$ It obviously converges for any $x\ $ of the form $\ 2^mk \pi\ $ where $\ m,k\in\mathbb{Z},\ $ but for any other values of $\ x\ $ the question is interesting…
Adam Rubinson
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Series convergent but not absolutely? $\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$

For which real numbers $p>0$ does the series $$\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$$ converge? Obviously it converges absolutely for $p>1$ but what about $0
Georgy
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Associativity of infinite products

It is well-known that if $\sum_{n=1}^\infty a_n$ is an absolutely convergent complex series and $\mathbb N$ is partitioned as $J_1,J_2,\dots$, then the series $\sum_{j\in J_n}a_j$ for all $n$ and $\sum_{n=1}^\infty\sum_{j\in J_n}a_j$ are both…
St. Barth
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Is there a sequence so that $\sum |a_n|=\infty$ and $\sum a_n \cos(nx)$ and $\sum a_n \sin(nx)$ converge everywhere?

Let the series be $(s_n a_n)_n$ where $s_n\in\{-1,1\}$ and $a_n$ decreases to $0$ instead for convenience. If $s_n$ is eventually periodic you can choose an $x$ which is a rational multiple of $\pi$ and most of the terms $s_n\cos(nx)$ are…
Derivative
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Group of all permutations of $\mathbb N$ that don't change the limit of series

Clearly all bijections functions $\varphi : \mathbb N \to \mathbb N$ form a group, let's call it $S(\mathbb N)$. Now let's define $H \subset S(\mathbb N)$ to be the set of all elements $\varphi \in S(\mathbb N)$ such that for every real valued…
principal-ideal-domain
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For any conditionally convergent series $\sum _{n=1}^\infty a_n,\ \exists\ k\geq 2\ $ such that the subseries $\sum _{n=1}^\infty a_{nk}$ converges.

A subseries of the series $\displaystyle\sum _{n=1}^\infty a_n$ is defined to be a series of the form $\displaystyle\sum _{k=1}^\infty a_{n_k}$, for $n_k \subseteq \Bbb N$. Prove or disprove: For any conditionally convergent series…
Adam Rubinson
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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 $$ 0 \le \left|\frac{\cos(n)}{2}\right| \le \frac{1}{2} \implies \lim_{n\to\infty}\left|\frac{\cos(n)}{2}\right| \le \frac{1}{2} < 1. $$ By…
Htamstudent
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