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        riddles >> putnam exam (pure math) >> Must this series converge?
        
(Message started by: ecoist on Apr 15^th, 2006, 8:58am)

Title: Must this series converge?
Post by ecoist on Apr 15^th, 2006, 8:58am

Let {a_n} and {b_n} be monotone increasing unbounded sequences of positive real numbers. Must the series

[sum]a_n^-b_n

always converge?

Title: Re: Must this series converge?
Post by Barukh on Apr 20^th, 2006, 9:15am

If I understand correctly, the answer is no: take b_n = log log(n), and a_n^b_n = n. For an appropriate base, both sequences are strictly increasing.

Title: Re: Must this series converge?
Post by ecoist on Apr 20^th, 2006, 2:22pm

Here's a different, more specific solution. Let b_n=sqr(ln n) and a_n=e^b_n, for n>1. Then a_n^-b_n=1/n, and the series is the divergent harmonic series (as is Barukh's solution).