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If that is what Mathematica thinks it sums to then you can be reasonably confident that there is no simple closed form for that sum in terms of elementary functions.View attachment 145 where M>=2. A close upper bound also will be useful(not like 1 as the upper bound). Thanks in advance.
This is also QPochhammer[1/M,1/M,inf]. Courtesy to mathematica.
The function...View attachment 145 where M>=2. A close upper bound also will be useful(not like 1 as the upper bound). Thanks in advance.
This is also QPochhammer[1/M,1/M,inf]. Courtesy to mathematica.
You can get it bounded away from 0 (for any $m>1$) like this. First, for $0<x<1$, $$ -\ln(1-x) = x + \tfrac{x^2}2 + \tfrac{x^3}3 + \ldots < x + x^2 + x^3 + \ldots = \tfrac x{1-x}.$$Can you tell anything about the lower bound? My doubt is whether it will converge to zero or not?