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.
**Hint** Let $S\_n$ be the partial sum of $\sum a\_n$ and $T\_n$ be the partial sum of $\sum (a\_n+a\_{n+1})$. Then 
$$T\_n-2S\_n =a\_{n+1}-a\_1$$

Since $T\_n-2S\_n$ is convergent to $-a\_1$, it follows that $T\_n$ is convergent if and only if $S\_n$ is convergent [this claim is an immediate consequence of the fact that the sum/difference of two convergent sequences is convergent].