$\mathrm{If} \ \ \sum_{k=1}^{\infty }a_{k} \ \ \mathrm{converges, is} \ \ \sum_{k=1}^{\infty }\sin a_{k} \ \ \mathrm{convergent?} $

Asked Mar 31 '25 at 08:24

Active Mar 31 '25 at 08:24

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$ \mathrm{The\ case\ } \ a_{k} > 0 \ \ \mathrm{is\ easy\ to \ proof\ since\ } \ \ |\sin(a_{k})|<a_{k}\ . \\ \ \mathrm{However,\ I\ found\ it\ hard\ to\ find\ a\ counter\ example\ without\ the\ condition. } $

asked Mar 31 '25 at 08:24

tka

1

I can't recall the exact reference at this moment, but we have the following result: Suppose $f:\mathbb{R}\to\mathbb{R}$ is such that $\sum_{k=1}^{\infty}f(a_k)$ converges whenever $\sum_{k=1}^{\infty}a_k$ converges. Then there exist constants $\delta > 0$ and $c\in\mathbb{R}$ such that $f(x)=cx$ for all $x\in(-\delta,\delta)$. That is, such $f$ is essentially a linear transformation. – Sangchul Lee Mar 31 '25 at 08:30

$\mathrm{If} \ \ \sum_{k=1}^{\infty }a_{k} \ \ \mathrm{converges, is} \ \ \sum_{k=1}^{\infty }\sin a_{k} \ \ \mathrm{convergent?} $

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