# Confused About Step in Proof of Divergence of $\sum \frac{1}{p}$

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.

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

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