# Questions tagged [absolute-convergence]

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

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### Absolute and conditional convergence of a non-alternating series

I tried all tests such as d'Alembert test, Cauchy test, Leibniz test,... but i could't determine convergence of this series: $$\sum_{n=2}^{\infty }\frac{(-1)^n}{\sqrt{n}+(-1)^n}$$ Can you help me!!!
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### Convergence of series from inverse of Cauchy product

The Cauchy product of two real or complex infinite series $\sum_{n\in\mathbb{N}} a_n$ and $\sum_{n\in\mathbb{N}} b_n$ is defined as: $$\forall n\in\mathbb{N}, c_n = \sum_{k=0}^n a_k b_{n-k}$$ ...
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### How to Prove the Divergence of an Improper Integral Involving Absolute Value

I'm working on understanding the convergence properties of certain improper integrals and encountered the following integral: $$\int_{0}^{\infty} \left| \frac{\cos(x)}{\sqrt{x}} \right| \, dx$$ I ...
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### Rearranging conditionally convergent series without changing the limit

Let $\{a_n\}_{n\in \mathbb{N}}$ be a sequence of real numbers such that the series $\sum a_n$ is conditionally convergent, i.e. the limit $\lim\limits_{N\to \infty} \sum_{n=0}^Na_n =:L \in \mathbb{R}$ ...
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### Convergence of $\sum |a_n|$

I have question about convergence of $\sum |a_n|$, $1.$ If $\sum |a_n|$ converges then $\sum a_n^k$ converges of any $K \in N$, my reasoning was, an $\sum |a_n|$ converges which means $\exists N$ ...
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### If the absolute value of an infinite series is bounded above, then the infinite series absolutely converges?

I am attempting to follow this simple example found in Further Linear Algebra by Blyth & Robertson page 14. It deduces that a sequence of partial sums must be absolutely convergent when bounded ...
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### Integrability of the Jacobi Theta Function

Let $$\psi(x) = \sum_{n = 1}^{\infty} e^{-n^{2} \pi x}$$ be a theta function. Can it be shown that that $$\int_{0}^{\infty} \psi(x) \cdot dx < \infty$$ without invoking Fubini-Tonelli’s Theorem ...
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### Absolute Convergence of a Series (maybe use ratio test) [closed]

Let $p:=\lim_{{k \to \infty}} k(1-|\frac{a_{k+1}}{a_k}|)$ Prove $p>1$ or $p=\infty \implies \sum_{n=1}^\infty a_n$ converge absolutely
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### What happens to EX if E|X| is infinity?

---------original question---------------- According to my professor, we can divide $X$ into $X_+=\max(X,0)$ and $X_-=-\min(X,0)$, both nonnegative. And we have $EX=EX_+-EX_-$ and $E|X|=EX_++EX_-$ For ...
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### At which value of K, the sum converges

I have a infinite sum that looks like the following: $$\sum_{k = 0}^{\infty} \left(\frac{1}{1 + k/A}\right)^a \frac{(-i B)^k}{k!}$$ Here, A is a positive real number and $a$ is a positive integer. ...
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### What happens if I change the norm of an absolute convergent series?

Let us consider the topological vector space $\mathbb{C}$ equipped with the euclidean topology. We know that a series of complex numbers $\sum_{n=1}^{\infty}a_n$ is said to be absolutely convergent ...
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### Question about proof of the truncated Perron's formula dealing with bounds and convergence

I have a question about the proof of the truncated Perron formula in my analytic number theory lecture notes. The Formula is given as follows: Let $x,c,T>0$ and suppose that $\sum_n |a_n|/n^c$ is ...
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