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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.

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    $\begingroup$ Yes. Actually the sum over all Mersenne numbers M:k, prime or not, converges. $\endgroup$
    – user23365
    Sep 17, 2012 at 8:00

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

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