Subsequence Convergence

**renolovexoxo** · January 31st 2013, 11:26 AM

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?

**Plato** · January 31st 2013, 12:15 PM

Originally Posted by renolovexoxo

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.

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 ${S - \varepsilon < S_N \leqslant S}$ .

**renolovexoxo** · January 31st 2013, 01:04 PM

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.

**Plato** · January 31st 2013, 01:29 PM

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.

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