Question

Whether the series converges or diverges

Answer 1

For
`k\ge1`, you have
`\sin\frac\pi k\le1`, and so


`\frac1{k^5}\sin\frac\pi k\le\frac1{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.