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Given the series: $$S=\sum_{k=1}^{N}\frac{k}{p_k}$$ where $p_k$ is the $k^{th}$ prime number, is it possible to know if this series converges in the limit: $$\lim_{N\to\infty}S$$ and eventually, its numerical value? Thanks in advance.

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    $\begingroup$ The sum diverges rather quickly, with $S \approx N/\log N$. Did you have a reason to expect it to converge? $\endgroup$ – Erick Wong Apr 9 '13 at 14:28
  • $\begingroup$ @Erick Wong: No, I don't. So the answer seems to be very simple. $\endgroup$ – Riccardo.Alestra Apr 9 '13 at 14:40
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The sum of inverse primes itself is divergent (see Wikipedia), so no the series does not converge.

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From the prime number theorem $$ \pi(x) = \frac{x}{\log x} + O\left(\frac{x}{(\log x)^2}\right) \qquad (x \to \infty) $$ one can deduce that $$ p_k \sim k \log k \qquad (k \to \infty), $$ which allows you to employ the limit comparison test on your sum.

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