I like to present convergent telescoping identities as infinite sums, e.g., inf ==== \ x x tan(x) = > (2 csc(------) - csc(--)), / n - 1 n ==== 2 2 n = 0 with a parameter (here x), two instances of which can be subtracted to give the sum over any range. But this example immediately suggests k - 1 ==== x \ x cot(--) - cot(x) = > csc(--), k / n 2 ==== 2 n = 0 which blows up in one direction and goes nuts in the other. Integrating, inf ==== | n | \ log(|cot(2 x)|) > ---------------- = - 2 (log(abs(sin(x))) + log(2)), / n ==== 2 n = 0 sort of reenabling the infinite series convention, but the lhs is undefined for x = dyadic rational * pi, a dense set! There may also be x for which it doesn't converge. Further massaging finally gives the very convergent product, 2 x %e coth(x) sqrt(coth(2 x)) sqrt(sqrt(coth(4 x))) . . . = ---------- . 2 4 sinh (x) --rwg