Articles of absolute convergence

Normed space where all absolutely convergent series converge is Banach

This question already has an answer here: If every absolutely convergent series is convergent then $X$ is Banach 2 answers

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.

Improper integral $\int_0^\infty \frac{\sin(x)}{x}dx$ – Showing convergence.

1)Show that for all $n\in\mathbb{N}$ the following is true: $$\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$$ for a constant $C>0$ and conclude that the improper integral $\int_0^\infty \frac{\sin(x)}{x}dx$ isn’t absolutely convergent. 2)Show that the improper integral $\int_0^\infty \frac{1-\cos(x)}{x^2}dx$ is absolutely convergent. (The integrand is to be expanded continuous at $x=0$.). 3)Using 2), show that the improper integral $\int_0^\infty \frac{\sin(x)}{x}dx$ […]

How to decide if it the series absolute convergent or conditional?

$$\sum_{n = 1}^{\infty}(-1)^{\frac{n(n+1)}{2}}(n^{\frac{1}{\sqrt[10]{n+10}}}-1)$$ Actually, I don’t have idea how to do it. Help me please Thanks a lot.

$f$ such that $\int_1^{\infty}f(x)dx$ converges, but not absolutely?

What’s an easy example of a function $f$ such that $$\int_1^{\infty}f(x)dx$$ converges, but not absolutely?

Prove that $\sum\limits_{n=0}^{\infty}{(e^{b_n}-1)}$ converges, given that $\sum\limits_{n=0}^{\infty}{b_n}$ converges absolutely.

It’s a question from a test that I had, and I don’t know how to prove this, so I am forwarding this to you. $\sum \limits_{n=0}^{\infty }\:b_n$ is absolutely convergent series . How to prove that the series $\sum \limits_{n=0}^{\infty }\:(e^{b_n}\:-\:1)$ is also absolutely convergent? We cannot assume here that $b_n\:\ge 0$ for every $n$.

Two problems on real number series

Consider the series: $$a) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\sin\frac{1}{2k})}$$ $$b) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\tan\frac{3}{2k})}$$ Showing that these two are convergent (and absolutely convergent) it’s no big deal for $a\in R^*$. But I couldn’t figure out how to solve it when $a=1$. My guess is that the general term has to be brought to another […]

Splitting an infinite unordered sum (both directions)

This question is a follow-up to this. Let $A, B_1, B_2, \dots$ be countable sets such that $\bigcup_{i \in \mathbb{N}}B_i = A$ and the $B_i$’s are disjoint. Let $f : A \to \mathbb{R}$ take elements of $A$ to the reals. The claim is that $$\sum_{w \in A} f(w) = \sum_{i \in \mathbb{N}} \sum_{w \in B_i} […]

Show that $\sum_{n = 1}^\infty n^qx^n$ is absolutely convergent, and that $\lim_{n \rightarrow \infty}$ $n^qx^n = 0$

I’m having trouble with proving the following for my math study: Let $x$ be a real number with $|x| < 1$, and $q$ be a real number. Show that the series $\sum_{n = 1}^\infty n^qx^n$ is absolutely convergent, and that $\lim_{n \rightarrow \infty}$ $n^qx^n = 0$ I tried this: $\sum_{n = 1}^\infty n^qx^n$ = $x […]

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 $\sum _{n=1}^\infty a_{n_k}$ converges. Suggested solution: $\Rightarrow$ We assume $\sum _{n=1}^\infty a_n$ converges absolutely $\Rightarrow \lim_{n […]