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Let $\{a_n\}$ be a sequence and $\{a_{n_k}\}$ be a subsequence. I have been asked to consider the relationships between the convergence of the series, \begin{align*} \sum_{i=1}^\infty a_n,\\ \sum_{i=1}^\infty a_{n_i}. \end{align*}

For terminology we are calling these the "series" and "subseries" respectively. In the reverse direction assuming that I have a convergent subseries I can show that the series does not necessarily converge using the harmonic series as the counterexample. We know that the harmonic series does not converge but if we only take the terms of the form $1/n^2$ which it contains as terms then the subseries converges by the p-test.

I am having difficulty in the other direction. If I have a convergent series I can't seem to work out if all subseries also converge. I'm not sure how to go about trying to prove it because when we start to remove elements from the sequence the terms in the sequence of partial sums change. On the other hand I also can't come up with a reasonable counterexample. Any help would be greatly appreciated.

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$\sum \frac {(-1)^{n}} n$ is convergent but the subseries formed by even numbered terms is divergent.

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As shown in another answer, there are convergent series with divergent subseries. But you can show, that for an absolutely convergent series every subseries is also absolutely convergent.

Proof: If $\sum_k a_k$ is absolutely convergent and $\sum_k a_{\pi(k)}$ an abritrary subseries of $\sum_k a_k$ with $\pi\colon\mathbb N\rightarrow\mathbb N$ strictly increasing, then you have for every $n\in\mathbb N_{\geq 1}$ $$\pi(0)<\pi(1)<\cdots<\pi(n-1)<\pi(n)$$ and therefore $$\sum_{k=0}^n |a_{\pi(a)}|\leq\sum_{k=0}^{\pi(n)}|a_k|\leq\sum_{k=0}^\infty |a_k|<\infty.$$ Hence, for $n\rightarrow\infty$ with $\pi(n)\rightarrow\infty$ you get the absolute convergence of the subseries $\sum_k a_{\pi(k)}$.

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