2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

@user3663006 , the logarithmic power series is

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

and from here

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

$\endgroup$
  • 1
    $\begingroup$ That makes sense, thanks for clearing that up. $\endgroup$ – user3663006 May 30 '14 at 19:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.