An Infinite Product: 10605
Jonathan Michael Borwein, Christopher G. Pinner, David M. Bradley · American Mathematical Monthly · 1999
Solution by David Bradley, University of Maine, Orono, ME. (a) First, for the case r = 1, the infinite product diverges to 0 because of the divergence of the harmonic series. Next consider the case r = 3. Let f (n) = n(n m)/(n2 mn + mi2). The product becomes Hln#Om f (n)/f (n + m). The product now telescopes, and since f (n) -+ 1 as n -+ oo, it reduces to f (2m) 11m -i7 f (n) and then to the given expression. (b) For each positive integer r, define fr(x) = Hln> (nr xr)/(nr + xr) when x is not an integer. Then for positive integers s and m, we have