Answer
FALSE
Work Step by Step
Given:$ \sum_{n=1}^{\infty}{n^{-sin1}}$ or $ \sum_{n=1}^{\infty}\frac{1}{n^{sin1}}$
Any series of the form $ \sum_{n=1}^{\infty}\frac{1}{n^{p}}$
is known as p-series.
A p-series is convergent if and only if $p>1$
Here $p=sin1$ and $sin1<1$
Thus, the series diverges.
Hence, the statement is false.