Given a sequence of n terms, a_{1}, a_{2}, ... , a_{n} the derived sequence is the sequence (a_{1}+a_{2})/2, (a_{2}+a_{3})/2, ... , (a_{n-1}+a_{n})/2 of n-1 terms. Thus the (n-1)th derivative has a single term. Show that if the original sequence is 1, 1/2, 1/3, ... , 1/n and the (n-1)th derivative is x, then x < 2/n.

**Solution**

By a trivial induction the n-1th derived sequence is the term (1/2^{n-1}) ∑ (n-1)Ci 1/i+1 = (1/n2^{n-1}) ∑_{0}^{n-1} nCi+1 = (2^{n}-1)/(n2^{n-1}) < 2/n.

© John Scholes

jscholes@kalva.demon.co.uk

8 Dec 2003

Last corrected/updated 8 Dec 03