# Questions tagged [absolute-convergence]

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

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### $\sum_{n=1}^\infty \int_{\mathbb R}|f_n| d\lambda<\infty \Rightarrow \sum_{n=1}^\infty f_n$ converges almost everywhere absolutely [closed]

Let $(f_n)_{n\in\mathbb N}$ be a sequence of measurable functions on $\mathbb R$ with $\sum_{n=1}^\infty \int_{\mathbb R}|f_n|d\lambda <\infty$. Show that $\sum_{n=1}^\infty f_n$ converges almost ...
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### Radius of convergence of power series when $x+y \leq n$

I can't figure out how to solve this problem... Any help would be thoroughly appreciated. Let $a_n$ be the number of pairs of $(x, y)$ such that $x + y \leq n$, where $x$ and $y$ are non-negative ...
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### Which conditions makes integrals of the form $\int_{-\infty}^\infty \left|\frac{\mathbb{P}(w^a)}{\mathbb{P}(w^b)} \right|dw < \infty$ converge?

Which conditions must fulfill an integral to be convergent $\int_{-\infty}^\infty \left|\frac{\mathbb{P}(w^a)}{\mathbb{P}(w^b)} \right|dw < \infty$ with $\mathbb{P}(w^m)$ a polynomial with ...
1 vote
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### Does $\sum^{∞}_{k=1} \frac{4k^k}{2^{k^2}}$ converge absolutely?

I was wondering if you guys could judge my reasoning and let me know if am correct in finding if this series converges absolutely or conditionally. $$\sum^{∞}_{k=1} \frac{4k^k}{2^{k^2}}$$ I decided to ...
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### Does series converge absolutely or conditionally?

I was wondering if you guys could judge my reasoning and let me know if am correct in finding if this series converges absolutely or conditionally. $$\sum^{∞}_{k=1} \frac{\sin(2k^2+1)}{k^{3/2}}$$ I ...
1 vote
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### Example of an absolutely summable series that is not summable

When I encountered Banach spaces I was presented with some proofs that link completeness, vector spaces, series and sequences (of partial sums). In particular I was presented with the following ...
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### Why Wolfram-Alpha (W-A) said that $\lim_{x \to \infty} e^{\sqrt{x}}\cdot\text{sinc}(x) = 0\,$?? [closed]

Why Wolfram-Alpha (W-A) said that $\lim_{x \to \infty} e^{\sqrt{x}}\cdot\text{sinc}(x) = 0\,$, if $\,\lim_{x \to \infty} \frac{e^{\sqrt{x}}}{x^n} = \infty$ for any $n\gg 0$?? I have tried the first ...
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### Kunita-Watanabe decomposition

proof Hello, I have a problem with this part of the proof to the Kunita-Watanabe decomposition. I don't understand how they use Lemma 1.69 to conclude that $\mathcal{G}$ is closed in the Hilbert space ...
68 views

### Convergence and absolute convergence of alternating series $\sum\int_{n\pi}^{(n+1)\pi} \frac{\sin t}{t}dt$

I have to determine the convergence (later absolute convergence) of the alternating series with the general term $$v_n=\int_{n\pi}^{(n+1)\pi} \frac{\sin t}{t}dt$$ using the usual alternating series ...
If I tell you that $S_n(x) = \sum^n_{k=0} \lvert f_k(x)\rvert$ converges uniformly, does it follow that $\lvert f_k(x)\rvert$ converge uniformly to zero? Assume that $f_k$ are real-valued functions ...
### Explanation of the proof of absolute convergence of $\zeta(s, a)$
Here is Theorem 12.1 of Introduction to Analytic Number Theory by Apostol - Theorem 12.1 The series for $\zeta(s, a)$ converges absolutely for $\sigma > 1$. The convergence is uniform in every ...