Re: Subsequence Convergence

Quote:

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 $\displaystyle a_n\ge 0$ then the sequence of partial sums is increasing.

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

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

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

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.

Re: Subsequence Convergence

Quote:

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: $\displaystyle \left( {S_n } \right) \to S$.

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