Question: prove or disprove 
if $\lim\_{n\to\infty} a\_n=0$ then $∑a\_n$ converges if and only if $∑a\_n+a\_{n+1}$ converges. 
I wasn't able to write the indices above and below the ∑. it should be from n=1 to infinity.

I've thought that this statement is incorrect at first, but I found no counterexample.

Answer: 
This is correct. Consider the partial sums for both series:
$$
S\_N = \sum\_{n = 1}^N a\_n
$$
and
$$
S\_N' = \sum\_{n = 1}^N (a\_n + a\_{n+1}) = 2 S\_N - a\_1 + a\_{N+1}
$$
Since $a\_n \to 0$, the sequence $S\_N$ converges if and only if the sequence $S\_N'$ converges.
Just consider the partial sums
$$ A\_N = \sum\_{n=1}^{N} a\_n,\qquad B\_N = \sum\_{n=1}^{N}\left(a\_n+a\_{n+1}\right) \tag{1}$$
and the relation
$$ B\_N = 2 A\_N - a\_1 + a\_{N+1} \tag{2}$$
together with the fact that $a\_{N+1}\to 0$ as $N\to +\infty$. In particular $\{A\_N\}\_{N\geq 1}$ is a convergent sequence *iff* $\{B\_N\}\_{N\geq 1}$ is a convergent sequence.