If $\sum_{n=1}^\infty x_n$ converges to s, and $y_n = \frac{x_n + x_{n+1}}{2}$ for all $n$, does $\sum_{n=1}^\infty y_n$ converge?

Question

if $\sum_{n=1}^\infty y_n$ does converge, what does it converge to?

I have already tried to rearrange $y_n$ but I don't know where to go from here or how the convergence of $x_{n+1}$ can be found.

Answer 1

Hint:$$\sum_{n=1}^\infty y_n=\frac{1}{2}\sum_{n=1}^\infty x_n+x_{n+1}=\frac{1}{2}\sum_{n=1}^\infty x_n+\frac{1}{2}\sum_{n=2}^\infty x_n$$

Knowing that $\sum_{n=1}^\infty x_n$ converges to $S$ what can we say about the last two sums?