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# Questions tagged [conditional-convergence]

This tag is for questions related to conditional convergence. A series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

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### Convergence of Riemann zeta function [duplicate]

I am wondering whether the series $$\zeta(s) = \sum_{n=1}^\infty n^{-s}$$ converges for $s$ with $\mathsf{Re}(s)=1$ and $\mathsf{Im}(s) \neq 0$. Note that I use the representation of $\zeta$ as an ...
• 1,793
1 vote
1 answer
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### Conditional convergent series implies existence of rearrangement that diverges: Doesn't the sum of the negative terms tend to $-\infty$?

In the proof for the Riemann Series Theorem that I'm reading, the author is currently establishing the existence of a divergent rearrangement of an infinite series given that the original series ...
• 669
2 votes
1 answer
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### how do you compute the value of $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$

I know that the series $\sum\limits_{n=1}^{\infty} \dfrac{(-1)^n}{4n-3}$ is convergent by Leibniz's law. However, finding the exact sum of this series can be quite challenging. I try to evaluate out ...
• 649
2 votes
0 answers
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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 $\lim\limits_{N\to \infty} \sum_{n=0}^Na_n =:L \in \mathbb{R}$ ...
• 952
3 votes
1 answer
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