Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent?

Question

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?

Answer 1

It doesn't. This is a famous result by Euler. An outstanding proof is found for example in Aigner, Ziegler "Proofs from THE BOOK".

Answer 2

The sum "behaves like" $\sum \frac{1}{n \log n}$ so that the sum of the first $n$ terms is approximately $ \log \log n$. The links to related questions in the first comment can tell more.

edit: I'll leave this up as it is non-duplicative of the linked material, but Will Jagy makes a convincing case that "consecutive" means "twin" and then it is the famous theorem by Brun (heuristically corresponding to convergence of $\int \frac{dn}{n (\log n)^2}$).

Answer 3

Your wording is ambiguous. Twin primes refer to pairs of primes $p,p+2.$ It is not known whether the collection of twin primes is finite or infinite. One reason it is not known is that the harmonic sum you write converges for the twin primes.

http://en.wikipedia.org/wiki/Twin_prime#Brun.27s_theorem

Answer 4

The series diverges. You will find multiple proofs here, including an especially compact and elegant proof due to Paul Erdos.