Find limit of $x_n = a ^ {x_{n-1}}$

Asked May 18 '23 at 08:09

Active May 18 '23 at 08:22

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We have $n > 1, a > 0, x_1 = 1$ and sequence $x_n = a ^ {x_{n-1}}$

Find $\lim_{n\to\infty} x_n$ for all values of $a.$

How can we solve this. Suppose we can prove that sequnce converges using Cauchy theorem for fundamental sequence. Also we should consider two possible ways: $a > 1$ and $a < 1.$ I think in the case of $a > 1$ sequence hasn't finit limit. But how to find exact limits? Any help is appreciated.

edited May 18 '23 at 08:22

Anne Bauval

34,650

asked May 18 '23 at 08:09

Angedon

4

Does this answer your question? Convergence of the sequence $x_{n+1} = a^{x_n}$ – Anne Bauval May 18 '23 at 08:21
Yes, partly, my task in addition requires finding all limits. – Angedon May 18 '23 at 08:27
2

To find the limit you have to use Lambert W-function, since it's $L=a^L$ – Zima May 18 '23 at 08:29

Find limit of $x_n = a ^ {x_{n-1}}$

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