The sequence a_{n} is monotonic and ∑ a_{n} converges. Show that ∑ n(a_{n} - a_{n+1}) converges.

**Solution**

The sum of the first n terms is (a_{1} - a_{2}) + 2(a_{2} - a_{3}) + ... + n(a_{n} - a_{n+1}) = a_{1} + a_{2} + a_{3} + ... + a_{n} - n a_{n+1}. We are given that ∑ a_{n} converges, so it is sufficient to show that the sequence n a_{n+1} converges to zero.

But since ∑ a_{n} converges, |a_{n+1} + a_{n+2} + ... + a_{2n}| is arbitrarily small for n sufficiently large. Since a_{n} is monotonic, this implies that n a_{n+1} is arbitrarily small for n sufficiently large.

© John Scholes

jscholes@kalva.demon.co.uk

5 Mar 2002