49th Putnam 1988

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Problem B4

αn are positive reals, and βn = αnn/(n+1). Show that if ∑ αn converges, then so does ∑ βn.

 

Solution

Note that βn > 2αn iff (1/αn)1/(n+1) > 2 or αn < 1/2n+1. So either βn < 2αn or βn < 1/2n. But both these series converge absolutely.

 


 

49th Putnam 1988

© John Scholes
jscholes@kalva.demon.co.uk
1 Jan 2001