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Prove the comparison test for infinite series: If $\vert a_i\vert \leq b_i$ for each $i$, and if the series $\Sigma b_i$ converges, then the series $\Sigma a_i$ converges. [Hint: show that the series $\Sigma\vert a_i \vert$ and $\Sigma c_i$ converge, where $c_i = \vert a_i \vert +a_i$.]

I came across this problem in Munkres, and it surprised me since this is a very analysis-like problem. I thought I'd give it a whirl, but I'm stuck. Here's what I have:

$\Sigma b_i$ converges $\Rightarrow \exists$ a sequence $b_n=\Sigma_{i=1}^nb_i$ such that $b_n\rightarrow b$. Define $s_n=\Sigma_{i=1}^n\vert a_i\vert$. Note that $s_n\leq s_{n+1}\forall n$. Then, $\vert a_i \vert\leq b_i $ for each $i$ $\Rightarrow$ $$\Sigma_{i=1}^n\vert a_i\vert\leq\Sigma_{i=1}^nb_i\\s_n\leq b\\$$ $\Rightarrow s_n$ is bounded above and increasing $\Rightarrow s_n\rightarrow s$ for some $s\in\mathbb R \Rightarrow \Sigma\vert a_i\vert$ converges.

I not sure how to show $\Sigma(\vert a_i\vert + a_i)$ converges. Maybe the answer is staring me in the face? I know once I get that, the rest is easy. Any insight? Thanks!

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  • $\begingroup$ I think you want to say there is a sequence $t_n=\sum_{i=1}^{n}b_i$ instead of $b_n=\sum_{i=1}^{n}b_i$, and that $t_n\rightarrow b$. $\endgroup$ – user84413 Sep 18 '14 at 0:35
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The sequence of partial sums $\sum_{i=1}^n (|a_i|+a_i)$ is non-decreasing and bounded above.

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