MathDB

Let

(x_n)_{n\geq 1}

be the sequence defined by

x_1\in (0,1)

and

x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}

for all

n\geq 1

. Find the values of

\alpha\in\mathbb{R}

for which the series

\sum_{n=1}^{\infty}x_n^{\alpha}

is convergent.

P1