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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.

718 questions
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Investigate the absolute convergence of the integral

I want to investigate the absolute convergence of integral. $$\int_{0}^{\infty} \; x^4 \; \sin(e^{2x}) \; dx$$ I made a replacement $$t = e^{2x},\; x = \frac{\ln{t}}{2} \\ \int_{1}^{\infty} \; \frac{\ln^4{t}}{32t} \; \sin{t} \;dt$$ I do not know…
Antoxa Barin
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How to prove the convergence of the cross entropy?

Cross entropy (CE) method is a probabilistic model based method. How to show that it always converges to an exact global optimal solution? How to set the number of samples $N$ and the number of elite samples $N_{\text{elite}}$?
Wenqi Zhou
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Convergence when ratio test=1

When using the ratio test for absolute convergence of a series $\sum_{n=1}^\infty a_{n}$, if the limit of the ratio $$|a_{n+1}|/|a_{n}|=1$$ when $n \rightarrow \infty$, the fate of the series is indeterminate. However, if $$|a_{n+1}|\ge|a_{n}|$$…
user460753
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Help with the AST

Can someone help me verify if this is true? When using the Alternating Series Test (AST), do I need to look at the absolute values of the terms and see if they converge to confirm that the series is absolutely convergent? If they don't converge,…
Jee
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How to prove this cosine series is not absolutely convergent?

I think the series $\sum_{k=0}^{\infty}\vert\cos(ak)\vert$, where $a$ is a non-zero constant, is absolutely convergent. But I had a trouble proving it. Can anyone give me a hint on how to prove it? Thanks!
SemiMetrics
  • 61