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IMC 2012 Problem 7

Define the sequence ${a_0,a_1,..}$ inductively by ${a_0=1, a_1=1/2}$ and

$\displaystyle a_{n+1}=\frac{na_n^2}{1+(n+1)a_n} \text{ for }n \geq 1.$

Show that the series ${\displaystyle \sum_{k=0}^\infty \frac{a_{k+1}}{a_k}}$ converges and determine its value.