If $a_n\to0,$ then $\sum a_n$ and $\sum (a_n + a_{n+1})$ converge/diverge together? Let $a_n$, a sequence suh that $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=0$
and the series: $\sum a_n$, $\sum (a_n + a_{n+1})$
Prove/Disprove: The series converge/diverge together.
I'll be glad for an hint or a guidance.

Hint: Consider $a_n = (-1)^n + 1/n^2$
