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In this post, the first answer states that a countably infinite summation can always be taken as an integral. Why is this? Is there some general formula that can turn any countably infinite summation into an integral?

ManUtdBloke
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  • How far do you know in measure theory ? – nicomezi Sep 02 '17 at 07:16
  • provided the series converges, otherwise there's not much point to it – AlvinL Sep 02 '17 at 07:20
  • @nicomezi I am currently on additivity for step functions and the lebesgue monotone convergence theorem. I did study measure theory a few years ago but didn't use it in the meantime so I remember little of it, hence why I'm going back through it rigorously now. – ManUtdBloke Sep 02 '17 at 07:24
  • @AlvinLepik Assuming the series converges then, what does this conversion operation look like? – ManUtdBloke Sep 02 '17 at 07:25

1 Answers1

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If the sum $\sum a_n$ is absolutely convergent, then yes: you can view the sequence $(a_0,a_1,\cdots)$ as a function $f$ on the measure space $(\Bbb N,\mu)$, where $\mu$ is the counting measure, and $\sum a_n$ as the integral $\int_\Bbb N f\,\mathrm d\mu$.

Vim
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