49th Putnam 1988

α_n are positive reals, and β_n = α_n^n/(n+1). Show that if ∑ α_n converges, then so does ∑ β_n.

Solution

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

49th Putnam 1988

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