Series prove or disprove statement 
Let $a_n$ be a sequence.
$\lim_{n\to\infty} a_n =0 $ then the series: $\sum_{n=1}^\infty a_n, \ \sum_{n=1}^\infty (a_n+a_{n+1}) $ converge and diverge together.

It doesn't seem right, there are no convergence tests that fit this question yet every counterexample I try does not seem to work.
Is this how I should see the sum?
$\sum_{n=1}^\infty (a_n+a_{n+1})=2\sum_{n=1}^\infty a_n +a_{n+1}$
If so then there's no real difference between the two series since you multiply it by a constant and add just one more value.

Recall that the series $\displaystyle\sum_{n\ge1}a_n$ is convergent if and only if the sequence
$\displaystyle\left(\sum_{k=1}^na_k\right)_n$ is convergent.
Now by this equality
$$\sum_{k=1}^n(a_{k+1}+a_k)-\sum_{k=1}^na_k=\sum_{k=1}^na_{k+1}=\sum_{k=1}^na_{k}-a_1+a_{n+1}$$
and since $\displaystyle\lim_n a_n=0$ we can deduce the desired result.
