Question

Whether the series converges or diverges

Answer 1

For $k\ge1$, you have $\sin\dfrac\pi k\le1$, and so

$\dfrac1{k^5}\sin\dfrac\pi k\le\dfrac1{k^5}$

If you're familiar with $p$-series, you know that

$\displaystyle\sum_{k\ge1}\frac1{k^p}$

converges for $p>1$. Here we have $p=5$. So the given series converges by comparison, and since it converges, it must converge absolutely.