Analysis(convergence of series) If partial sum has a convergent subsequence with lim a_n=0, then series converges?

Question

I have a question. I’m solving a problem;

Let <a_n>, <b_n> be two sequence in R. b_k=sum(a_n) from n=2^k-1 to 2(2^k-1) for each k in N. Show that if sigma(a_n)=1, then lim a_n=0 and sigma(b_n)=1. Dose the converse hold?

I solved a direction(=>). But I don’t know the converse is true or false.

My Attempt:

S_n: partial sum of a_n T_n: partial sum of b_n

Then T_n = S_2(2^n-1),i.e., T_n is a subsequence of S_n.

If lim a_n=0 and S_n has a convergent subsequence(T_n), then S_n converges??

If so, the converse will be true. Or any counter example??

Help me please:D