[math-fun] peculiar product constant

In[103]:= Product[Sin[n π/(2 n + 1)], {n, ∞}]==0.7435726169400367135862 During evaluation of In[103]:= Product::div: Product does not converge. And, as usual, I am the Virgin Mary. No joy from ries (now with Lambert-W) nor ISC nor Finch. The (log of the) slows-to-a-crawl product converts to a ~1 digit/term sum: Sum[(4^k - 1 )*BernoulliB[2*k]*((1 - 4^k)*π^(4*k)*BernoulliB[2*k] - 2*(-1)^k*π^(2*k)*(2*k)!)/(4*k*(2*k)!^2), {k,∞}] =-0.2962888487994070997 (How often do you see squared Bernoulli numbers?) FullSimplify splits this into two divergent series, one of which it formally recognizes as a simple constant, while the Bernoulli² sum just sits there smirking while approaching a harmonic series. --rwg