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I was going through the number theory text by Ireland and Rosen, and was following the proof of the divergence of the sum of reciprocal primes. But I came across a step unclear to me.

The proof so far: https://imgur.com/XbPkBak&JeMOqku

The step itself: https://imgur.com/XbPkBak&JeMOqku#1

I cannot see the equality between the two sides. Help is appreciated.

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  • $\begingroup$ They are using the taylor series for $\log(1-x)$. $\endgroup$ – RKD May 30 '14 at 19:52
  • $\begingroup$ Isn't it just Taylor series expansion? $\endgroup$ – Marcin Łoś May 30 '14 at 19:53
  • $\begingroup$ Ah, I didn't know about the expansion, thanks. $\endgroup$ – user3663006 May 30 '14 at 20:00
  • $\begingroup$ How does the proof finish? $\endgroup$ – abnry May 30 '14 at 20:10
  • $\begingroup$ Velcome to the site! $\endgroup$ – kjetil b halvorsen May 30 '14 at 20:10
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@user3663006 , the logarithmic power series is

$$\log(1-x)=-\sum_{n=1}^\infty\frac{x^n}n\;\;,\;\;|x|<1$$

and from here

$$\log(1-p^{-s})=-\sum_{n=1}^\infty\frac{p^{-ns}}n\;\ldots$$

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    $\begingroup$ That makes sense, thanks for clearing that up. $\endgroup$ – user3663006 May 30 '14 at 19:59

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