Convergent Series and Rearrangements

Question

Given a convergent positive series, I have to prove that $$\sum^{\infty}_{n=1} a_n=\sum^{\infty}_{k=0}a_{2k+1} + \sum^{\infty}_{n=1} a_{2k}$$ which means that the sum of odd terms and the even terms converges to the same series on the LHS. I know that a convergent positive series is absolutely convergent and I can use theorems on rearrangements (I think) but how can I write the logic of the analysis out?

Answer 1

Let $S_n=\sum_{k=1}^na_k$, $O_n=\sum_{k=1}^na_{2k+1}$, and $E_n=\sum_{k=1}^na_{2k}$. Then $$ 0\le O_n\le S_{2n+1}\le \sum_{k=1}^\infty a_k,\quad 0\le E_n\le S_{2n}\le \sum_{k=1}^\infty a_k,\quad O_{n}+E_n=S_{2n+1}. $$ The first two inequalities show that $O_n$ and $E_n$ are bounded. Since they are also increasing, both $O_n$ and $E_n$ converge. Take limits in the third equation to get the equality in your question.