The series $\displaystyle\sum^\infty_{n=1}a_n$ converges absolutely.
Prove that for every polynomial $p(x)=p_0+p_1x+...+p_dx^d$, $\ \displaystyle\sum^\infty_{n=1}p(a_n)$ converges iff $p_0=0$.
What I did:
$\displaystyle\sum^\infty_{n=1}p(a_n)=\displaystyle\sum^\infty_{n=1}(p_0+...p_dx^d)=\sum^\infty_{n=1}p_0+\sum^\infty_{n=1}\sum^d_{i=1}p_i(a_n)^i$
Let's take a look at $\displaystyle\sum^\infty_{n=1}\sum^d_{i=1}p_i(a_n)^i$, we know $a_n$ converges, raising it to an exponent won't change it's convergence and scaler multiplication won't either, so all 'elements' in the sum except for $p_0$ converge.
$\displaystyle\sum^\infty_{n=1}p_0$ will always diverge unless $p_0 = 0$.