# Sum involving prime numbers

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.

• The sum diverges rather quickly, with $S \approx N/\log N$. Did you have a reason to expect it to converge? – Erick Wong Apr 9 '13 at 14:28
• @Erick Wong: No, I don't. So the answer seems to be very simple. – Riccardo.Alestra Apr 9 '13 at 14:40

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.