Series of Mersenne primes

Question

If the 'Lenstra - Pomerance - Wagstaff' conjecture is true, there are infinite Mersenne primes. In this case, if we consider the series: $$S_N=\sum_{k=1}^N \frac{1 }{M_k}$$ where $M_k$ is $k^{th}$ Mersenne prime, does the limit: $$S_\infty=\lim_{N\to\infty}S_N$$ converges to a finite value? Thanks.

Answer 1

Yes, since \[ \sum_{k=1}^\infty \frac 1{M_k} \le \sum_{k=1}^\infty \frac 1{2^k-1} < \infty. \]