26th Putnam 1965

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Problem A3

{ar} is an infinite sequence of real numbers. Let bn = 1/n ∑1n exp(i ar). Prove that b1, b2, b3, b4, ... converges to k iff b1, b4, b9, b16, ... converges to k.

 

Solution

If a sequence converges to k, then any subsequence also converges to k.

So suppose that the square terms converge to k. Let cr = exp(i ar). Take two consecutive squares N = n2 and N' = n2 + 2n + 1 and m between them. Then |bN - bm| ≤ |bN - 1/N ∑ cr| + |(1/N - 1/m) ∑ cr |, where the sums are over m terms. But |bN - 1/N ∑ cr| ≤ 1/N ∑N+1m |cr| = (m - N)/N ≤ 2n/n2 = 2/n. Also |(1/N - 1/m) ∑ cr | ≤ (1/N - 1/m) ∑ |cr| < (1/n2 - 1/(n+1)2) (n+1)2 = (2n+1)/n2 < 3/n. So the difference |bm - bN| is arbitrarily small for n sufficiently large. Thus if we take m sufficiently large, then |bm - bN| < ε and |bN - k| < ε. So |bm - k| < 2ε and the sequence converges.

 


 

26th Putnam 1965

© John Scholes
jscholes@kalva.demon.co.uk
25 Jan 2002