For each positive integer n let an 1 n 1 1 n 1n 2 1 n

For each positive integer n, let

a_n =1 / (n + 1) +1/ (n + 1)(n + 2) + 1/ (n + 1)(n + 2)(n + 3) +

Solution

Although the series for bn does not at rst sight look tractable, it is in fact just a geometric progression: the rst term is 1/n + 1 and the common ratio is also 1/n + 1. Thus

b_n =[1/( n + 1)]( 1/ (1 - 1/(n + 1))) = 1 / n