Counter example to disprove the statement"if the series of sums of each two adjacent terms converges, then the whole series converges"

Question

It is well known that if \begin{equation} S_{1}=\sum_{n=1}^{\infty} F_{n} \end{equation} converges, then \begin{equation} S_{2}=\sum_{n=1}^{\infty} F_{2n-1}+F_{2n} \end{equation} converges

But I know that given that $S2$ converges then $S1$ may diverges.

Why is that true? Could someone give me a counterexample to let me see the fact? Or could someone give me a hint to build up such an counterexample? Many thanks!

Answer 1

Consider $F_n=(-1)^n$. Then we have

$$S_2=\sum_{n=1}^\infty(-1)^{2n-1}+(-1)^{2n}=0+0+0+\dots0=0$$

On the other hand,

$$S_1=\sum_{n=1}^\infty(-1)^n=-1+1-1+1-1+1-1+\dots$$

which doesn't converge to anything.