Question:
A subseries of the series $\sum _{n=1}^\infty a_n$ is defined to be a series of the form $\sum _{n=1}^\infty a_{n_k}$, for $n_k \subseteq \Bbb N$. Prove that $\sum _{n=1}^\infty a_n$ converges absolutely if and only if each subseries $\sum _{n=1}^\infty a_{n_k}$ converges.
Suggested solution:
$\Rightarrow$
- We assume $\sum _{n=1}^\infty a_n$ converges absolutely $\Rightarrow \lim_{n \to \infty} |a_n|$=0. We know from the definition of series that it's actually the sequence of partial sums so $\sum _{n=1}^\infty a_n = S_n$. Therefore we can treat it like a sequence and say that it converges to L. Since $S_n$ converges to L, each of it's sub sequences also converge to L. Therefore $\sum _{n=1}^\infty a_{n_k} =S_{n_k} \to L$ as well.
$\Leftarrow$
- We assume each subseries converges. Specifically the sub series $\sum _{\Bbb N even} a_n = A$ $\sum _{\Bbb N odd} a_n = B$ . Since these two series comprise all of the naturals, then $\sum _{n=1}^\infty a_n =\sum _{\Bbb N even} a_n+\sum _{\Bbb N odd} a_n = A+B$ . Therefore it converges to A+B.
I would like to verify this proof, because I've given it a lot of thought, and still not 100 percent sure about it. Please hint me, and notify me about any mistakes.
Thanks.
Best Answer
The two suggested proofs are wrong
$\Rightarrow$
We have $$\sum_{k=1}^\infty|a_{n_k}|=0\times|a_1|+\cdots+0\times|a_{n_1-1}|+|a_{n_1}|+0\times|a_{n_1+1}|+\cdots\leq\sum_{n=1}^\infty|a_{n}|$$ hence the series $$\sum_{k=1}^\infty|a_{n_k}|$$ is convergent by comparaison.
$\Leftarrow$
Let denote by $(a_{n_k})$ the subsequence of $(a_n)$ by choosing all its positive terms and $(a_{n'_k})$ the subsequence of $(a_n)$ by choosing all its negative terms then the series $$\sum_{k=1}^\infty a_{n_k} \quad\text{and} \quad \sum_{k=1}^\infty a_{n'_k}$$ are absolutely convergent and its clear that $$\sum_{n=1}^\infty |a_{n}|=\sum_{k=1}^\infty a_{n_k}-\sum_{k=1}^\infty a_{n'_k}$$ hence the series $$\sum_{n=1}^\infty a_{n}$$ is absolutely convergent.