Suppose a_{k}>= 0 for all k in N. Prove that the series from 1 to inifity of a_{k} converges iff some subsequence {s_{nk}} of the sequence {s_{n}} of partial sums converges.
I've proved that if a_{k} converges then the subsequence will converge. I'm have some trouble going the reverse dirrection, proving that if the subsequence converges then a_{k} will converge. Can I just take {s_{nk}}={s_{n}} and then by our assumption, since the sequence of partial sums converges the series a_{k} will converge?