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Question Number 8661 by FilupSmith last updated on 20/Oct/16

p_{n} = n^{t h} prime number

p_{1} = 2, p_{2} = 3, p_{3} = 5, \dots

Does the following converge :

\underset{i = 1}{\sum^{\infty}} \frac{p_{i}}{p_{i + 1}}

Prove / disprove

Answered by prakash jain last updated on 21/Oct/16

a_{n} = \frac{p_{n}}{p_{n + 1}}

Bertrand prime number theorem states

that p_{n + 1} < 2 p_{n}

\lim_{n \to \infty} a_{n} > \lim_{n \to \infty} \frac{p_{n}}{2 p_{n}} = \frac{1}{2}

Since \lim_{n \to \infty} a_{n} \neq 0 . The series diverges .