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Recursive sequence has convergent power series

Source: SEEMOUS 2024, Problem 1

April 16, 2024
real analysiscollege contests

Problem Statement

Let (xn)n≥1(x_n)_{n\geq 1} be the sequence defined by x1∈(0,1)x_1\in (0,1) and xn+1=xn−xn2nx_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}} for all n≥1n\geq 1. Find the values of α∈R\alpha\in\mathbb{R} for which the series ∑n=1∞xnα\sum_{n=1}^{\infty}x_n^{\alpha} is convergent.
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