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Series convergence

Posted: Sun Apr 16, 2017 4:35 pm
by Riemann
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.

Re: Series convergence

Posted: Tue Apr 18, 2017 5:09 pm
by Papapetros Vaggelis
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.