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 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.
 A: 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.
A: 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.
A: Let 
$S_N = \sum_{n=1}^N a_n$
and 
$T_N = \sum_{n=1}^N (a_n+a_{n+1}).$
Can you find a formula for $S_N$ in terms of $T_N$ and vice versa?  (And can you see why doing this would prove the statement?)
A: 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].
