IMC 2012 Problem 7

Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$ , $a_1=\frac{1}{2}$ , and $$a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \forall n \ge 1.$$ Show that the series $\sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value.

Proposed by Christophe Debry, KU Leuven, Belgium.