How do I craft this constant to a specific accuracy?

Question

An earlier post was migrated to Physics.SE and we determined that if this converged, we would have a constant close to the mass of the proton in kilograms. This constant is constructed by summing the reciprocals of the products of all the numbers between consecutive squares.

Per this:

SumConvergence[10^-26/Pochhammer[m^2 + 1, 2 m], m]  

(* True *)

we have convergence and thus a constant. However, the following:

Sum[10^-26/Pochhammer[m^2 + 1, 2 m], {m, \[Infinity]}]  

outruns the precision.

My question: Is there a way to use this sum to create the constant to a specified accuracy?

Answer 1

Clearly, you can factor out the tiny portion of your constant, so:

NSum[1/Pochhammer[m^2 + 1, 2 m], {m, 1, ∞}, WorkingPrecision -> 50] 1*^-26
   1.6726218229590580987863882056891582636342622102204*10^-27

OTOH,

...products of all the numbers between consecutive squares.

Product[j, {j, (k - 1)^2, k^2}]
   Pochhammer[(-1 + k)^2, 2 k]

This doesn't resemble what you have.