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max a wanted so that \lim_{n\to \infty} x_n exists where x_n=a^{x_{n-1}} , a>1

Source: Tuymaada 1995 p2

April 27, 2019
limitLimit of sequencerecurrence relationmaximumalgebra

Problem Statement

Let x1=a,x2=ax1,...,xn=axn−1x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}} where a>1a>1. What is the maximum value of aa for which lim exists lim⁡n→∞xn\lim_{n\to \infty} x_n and what is this limit?
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