Interesting.

Per

https://primes.utm.edu/notes/faq/NextMersenne.html re Lenstra and Pomerance's conjecture, the geometric mean of the ratio of successive Mersenne exponents is expected to be R ~1.4757614.

Let r=1/R = 0.677616314.

I think without knowing at least certain bounds on the distribution that determines the series' terms, one can not prove whether a series converges or diverges.

We know r < 1 for the real Mersenne primes exponent series (known and unknown), because we will sort them into ascending order.

We know r > 0.

Under these conditions a geometric series sum will converge.

Summing the series for that, from the first term, as an infinite series,

beginning with 1/2, 1/2 (1+r+r^2+r^3+...) = 1/2 /(1-r) = 1.550946966...,

~7.1% larger than the sum of reciprocals of the known Mersenne primes' exponents to date, ~1.448181855...

Some might point to that difference as indication of "missed" primes. I think that's wrong.

It could be that a=1/2 is not the correct multiplier value for the series.

(Choosing to match terms at the first Mersenne prime exponent is arbitrary. Matching at the second or later instead is also arbitrary and yields a mildly better comparison in a few cases I tried.)

The geometric series sum proves nothing. It only shows that certain assumptions regarding the conjecture yield results somewhat consistent with the known data.

The reciprocals of known Mersenne primes' exponents do not constitute a geometric series.

Assuming that whatever Mersenne primes are discovered in the GIMPS search up to 10

^{9} are consistent with the conjecture, they are likely to add less than 10

^{-7} to the sum of reciprocals of known Mersenne primes' exponents.

The empirical data looks persuasive that the sum converges.