Here is something I thought up while I was bored and my, erm, fish were busy:

Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that $$p=\sum_{k=1}^\infty \frac1{\prod\limits_{m=1}^n (c_m k+w_m)}?$$ If so, how do you find these integers?

I got curious about this when I started playing around with telescopic series. As you know, the good thing about them is that they are easily evaluated through clever rewriting of the general term, after which there is much cancellation, leaving behind a few terms that add up to the desired result.

That got me wondering on whether a rational number always has a telescopic series representation. I don't really know how this might be proven (and I'm not that good at math), so I wish somebody could enlighten me. Thanks!

  • $\begingroup$ an example, $p=\frac37$, given here... $\endgroup$ – draks ... Aug 2 '12 at 9:27
  • $\begingroup$ @draks, I have seen that, but the numerators in that answer are $\neq 1$, while I am asking for a telescopic series whose general term has $1$ as the numerator. Otherwise, you can always just multiply whatever telescopic series with an appropriate rational factor. $\endgroup$ – Timmy Turner Aug 2 '12 at 9:30
  • $\begingroup$ ok, good luck and +1 $\endgroup$ – draks ... Aug 2 '12 at 9:34

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