1. ## Subsequence Convergence

Suppose ak>= 0 for all k in N. Prove that the series from 1 to inifity of ak converges iff some subsequence {snk} of the sequence {sn} of partial sums converges.

I've proved that if ak converges then the subsequence will converge. I'm have some trouble going the reverse dirrection, proving that if the subsequence converges then ak will converge. Can I just take {snk}={sn} and then by our assumption, since the sequence of partial sums converges the series ak will converge?

2. ## Re: Subsequence Convergence

Originally Posted by renolovexoxo
Suppose ak>= 0 for all k in N. Prove that the series from 1 to inifity of ak converges iff some subsequence {snk} of the sequence {sn} of partial sums converges.
Because $a_n\ge 0$ then the sequence of partial sums is increasing.

Now suppose that $\left( {S_{n_i } } \right) \to S$.

If $\varepsilon > 0$ then $\left( {\exists K} \right)\left[ {S - \varepsilon < S_{n_K } \leqslant S} \right]$.

But if $N\ge n_K$ you know that ${S - \varepsilon < S_N \leqslant S}$.

3. ## Re: Subsequence Convergence

Would you mind explaining the very last inequality just a little further for me? I follow you until that conclusion, then I'm a little lost.

4. ## Re: Subsequence Convergence

Originally Posted by renolovexoxo
Would you mind explaining the very last inequality just a little further for me? I follow you until that conclusion, then I'm a little lost.
In effect, it shows that the sequence of partial sums converges: $\left( {S_n } \right) \to S$.

Again that is because the sequence of partial sums is increasing.