Evaluate the following summation: Starting at n=1, ending at infinity. (1/(n(n+1)))

Answer 1

∑_n=1^∞1/(n(n + 1)) = S;

S_k = ∑_n=1^k1/(n(n + 1)) = ∑_n=1^k[1/n - 1/(n+1)] =1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ...+ 1/(k - 2) - 1/(k - 1) + 1/(k - 1) - 1/k + 1/k - 1/(k + 1) = 1 - 1/(k + 1).

S = lim k→∞ S_k = lim k →∞ (1 - 1/(k + 1)) = 1 - 0 = 1