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 Post subject: Series convergencePosted: Sun Apr 16, 2017 4:35 pm

Joined: Sat Nov 14, 2015 6:32 am
Posts: 137
Location: Melbourne, Australia
Let $\{a_n\}_{n \in \mathbb{N}}$ be a sequence of positive real number such that $\sum \limits_{n=1}^{\infty} a_n$ converges. Is the series

$$\mathcal{S} = \sum_{n=1}^{\infty} n a_n \sin \frac{1}{n}$$

also convergent? Give a brief explanation.

_________________
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$

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 Post subject: Re: Series convergencePosted: Tue Apr 18, 2017 5:09 pm
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Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
For every $\displaystyle{n\in\mathbb{N}}$ holds

$\displaystyle{\left|n\,a_n\,\sin\,\dfrac{1}{n}\right|=n\,a_n\,\left|\sin\,\dfrac{1}{n}\right|\leq n\,a_n\,\dfrac{1}{n}=a_n}$

and $\displaystyle{\sum_{n=1}^{\infty}a_n<\infty}$.

So, the series $\displaystyle{\sum_{n=1}^{\infty}n\,a_n\,\sin\,\dfrac{1}{n}}$ converges since

it converges absolutely.

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