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Convergence of series Brazilian MO 2022 P4

Source: Brazilian Mathematics Olympiad 2022 #4

November 22, 2022
seriesreal analysis

Problem Statement

Let α,c>0\alpha, c > 0, define x1=cx_1 = c and let xn+1=xne−xnαx_{n + 1} = x_n e^{-x_n^\alpha} for n≥1n \geq 1. For which values of β\beta does ∑i=1∞xnβ\sum_{i = 1}^{\infty} x_n^\beta converge?
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