p-n-nth-prime-p-1-2-p-2-3-p-3-5-Do-the-following-sums-converge-Prove-disprove-1-S-n-1-n-p-n-2-S-n-1-n-p-n-2-

Question Number 7723 by FilupSmith last updated on 12/Sep/16

p_{n} = n th prime (p_{1} = 2, p_{2} = 3, p_{3} = 5, \dots)

Do the following sums converge ? Prove / disprove .

(1) S = \underset{n = 1}{\sum^{\infty}} \frac{n}{p_{n}}

(2) S = \underset{n = 1}{\sum^{\infty}} \frac{n}{p_{n}^{2}}

Commented by FilupSmith last updated on 12/Sep/16

is the following correct for (1) :

\lim_{n \to \infty} \frac{n}{p_{n}} = \frac{\lim_{n \to \infty} n}{\lim_{n \to \infty} p_{n}} = \frac{\infty}{\infty}

L^{'} hopital

\frac{\frac{d}{d n} (n)}{\frac{d}{d n} (p_{n})}

= {(\frac{d}{d n} (p_{n}))}^{- 1}

= ? ? ?