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Putnam 2011 A2

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December 5, 2011

Putnamratiogeometric seriescollege contestsSummation

Problem Statement

Let

a_1,a_2,\dots

and

b_1,b_2,\dots

be sequences of positive real numbers such that

a_1=b_1=1

and

b_n=b_{n-1}a_n-2

for

n=2,3,\dots.

Assume that the sequence

(b_j)

is bounded. Prove that

S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}

converges, and evaluate

S.

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