# Questions tagged [absolute-convergence]

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

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### Show a sum in a Banach space is convergent and bounded

$f,g \in l^2(\mathbb{Z})$ such that $\sum_{n \in \mathbb{Z}} |f(n)|^2 < \infty, \sum_{n \in \mathbb{Z}} |g(n)|^2 < \infty$. Show that $(f \star g)(n) = \sum_{m \in \mathbb{Z}} f(m)g(-m+n)$ is ... 1 vote
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### Example for convergence and absolute convergence abscissa

Let $f\in L^1_{loc}(\mathbb{R})$ a locally integrable function, with $f:\mathbb{R}\rightarrow\mathbb{C}$. We say that $\int_0^{\infty} f(t) dt$ converges if the limit \begin{equation} \lim_{b\to\infty}...
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### Can I use comparison test in this situation?

When you are given only that $\sum a_k$ converges but you're not sure if it converges absolutely, and you need to prove some related series $\sum b_k$ also converges, should I probably not use ...
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### Why does $E|X|=\infty$ imply $EX=\infty$?

In Casella and Berger Example 2.2.4(Cauchy mean), in trying to show that Cauchy random variable $X$ has no mean, the authors prove that $E|X|=\infty$. Since we want to prove that $EX=\infty$, it has ...