αn are positive reals, and βn = αnn/(n+1). Show that if ∑ αn converges, then so does ∑ βn.
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
1 Jan 2001