I have question about convergence of $\sum |a_n|$,
$1.$ If $\sum |a_n|$ converges then $\sum a_n^k$ converges of any $K \in N$,
my reasoning was, an $\sum |a_n| $ converges which means $\exists N$ such that $\forall n > N, $ $|a_n| < 1$.,
so any $K \in N$, $a_n^k \leq |a_n|$ that's because of $|a_n|<1$ $\forall n>N$.
so $\sum_{N+1}^{\infty} a_n^k$ $\leq$ $ \sum_{N+1}^{\infty}|a_n|$, so by direct comparison test $\sum a_n^k$ converges.
correct me if i was wrong.