Given a sequence of n terms, a1, a2, ... , an the derived sequence is the sequence (a1+a2)/2, (a2+a3)/2, ... , (an-1+an)/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 (1/2n-1) ∑ (n-1)Ci 1/i+1 = (1/n2n-1) ∑0n-1 nCi+1 = (2n-1)/(n2n-1) < 2/n.
© John Scholes
jscholes@kalva.demon.co.uk
8 Dec 2003
Last corrected/updated 8 Dec 03