Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

420 questions
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$\sqrt{n}(x_n-x_0) \to N(0,\sigma_0^2)$ in distribution implies $x_n \to x_0$ almost surely

Is this true? I figured that if not, there will some positive probability $\sigma$ that $\sqrt{n}(x_n-x_0)$ takes $\sqrt{M} \cdot \epsilon$ for infinitely many large $M$. Even though this "blowing up" ...
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Find the range of convergence of the series$\,\,\sum_{n=0}^\infty {\frac{z^n}{1+z^{2n}}}$

The series I have is $$\displaystyle\sum_{n=0}^\infty {\dfrac{z^n}{1+z^{2n}}}$$ The same series with absolute values is: $$\displaystyle\sum_{n=0}^\infty {\dfrac{|z|^n}{1+|z|^{2n}}}$$ Using D'...
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Normed Space $X$ complete iff any absolutely comvergent series in $X$ converges

I'm studying functional analysis. I have trouble with the following proposition and its proof. Wonder if someone could help me with the following questions: Proposition: A normed space $X$ is ...
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Determine Convergence of $\sum_{n=1}^\infty \frac{(-1)^nn!}{(n+100)!}$

I know that $\frac{n!}{(n+1)!}$ can be reduced to $\frac{1}{n+1}$, but i'm not sure about this one. $$\sum_{n=1}^\infty \frac{(-1)^nn!}{(n+100)!}$$ In my notes, my professor reduced it to a p-...
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Why we ONLY use ratio test and not conditional convergence to determine the interval of convergence of an alternating series?

For example, consider $$S_n=\sum_{n=1}^{\infty} \frac{(-1)^n x^n} {\sqrt{n}}$$ While determining the interval of convergence, we use the ratio test to determine the interval in which the series ...
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Absolute convergence of the series $\sum_1^{\infty} {(-1)}^n(\sqrt{n+1} -\sqrt{n})$

$\sum_1^{\infty} {(-1)}^n(\sqrt{n+1} -\sqrt{n})$ I want to know if this series diverge or if it is absolutely convergent or conditionally convergent. I used Leibniz' Criterion of alternating series ...
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If $\sum\limits_{n=1}^∞u_n^2$ is convergent, then $\sum\limits_{n=1}^∞\frac{u_n}n$ is absolutely convergent [duplicate]

If $\{u_n\}$ is a sequence of real numbers and the series $\displaystyle\sum_{n=1}^{\infty}u_n^2$ is convergent, prove that the series $\displaystyle\sum_{n=1}^{\infty}\frac{u_n}{n}$ is absolutely ...
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Finding a sequence $a_n$ that diverges such that $\|a_n\|$ converges (in $\mathbb{R}$)

Finding a sequence $a_n$ that diverges such that $\|a_n\|$ converges (in $\mathbb{R}$) I am having a hard time finding an example that works. An example or hint would be greatly appreciated.
Let $\sum a_nx^n$ be a series, where $a_n$'s are not necessarily positive & let $R>0$ be its radius of absolute convergence. Then for $x=r>R$, $\sum |a_n|r^n$ will be divergent, but my ...