TITLE: if $\sum_{n=0}^\infty na_n$ converges, does $\sum_{n=0}^\infty na_{n+1}$ converge?
QUESTION [5 upvotes]: If the sum :$$\sum_{n=0}^\infty n\, a_n$$ converges does it mean that : $$\sum_{n=0}^\infty n\, a_{n+1}$$ converges too?
(Note that $a_n$ can be negative)

REPLY [2 votes]: Given that $\sum_{n\geq 1}n a_n$ is convergent, we want to show that $\sum_{n\geq 1}(n-1)a_n$ is convergent as well, so it is enough to show that in the given hypothesis $\sum_{n\geq 1}a_n$ is convergent. 
Let $A_n = 1 a_1 + 2 a_2 + \ldots + n a_n$. By summation by parts:
$$ \sum_{n=1}^{N} a_n = \sum_{n=1}^{N}\frac{1}{n}(na_n) = \frac{A_N}{N}+ \sum_{n=1}^{N-1} \frac{A_n}{n(n+1)} $$
and $A_n\to C$ as $n\to +\infty$, hence $\sum_{n\geq 1}\frac{A_n}{n(n+1)}$ is an absolutely convergent series and $\sum_{n\geq 1}a_n$ is convergent (at least conditionally).