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Recurrence and Series

Source: IMC 2012, Day 2, Problem 2

July 29, 2012
limitIMCcollege contests

Problem Statement

Define the sequence a0,a1,…a_0,a_1,\dots inductively by a0=1a_0=1, a1=12a_1=\frac{1}{2}, and a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n},   \forall n \ge 1. Show that the series ∑k=0∞ak+1ak\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k} converges and determine its value.
Proposed by Christophe Debry, KU Leuven, Belgium.
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