Nested Fractional Exponent Discovered

Calculus Level 3

The sequence \( (-1), {\large (-1)^ \frac{1}{(-1)} }, {\Large (-1) ^ \frac{1}{(-1)^ \frac{1}{(-1)}} }, \ldots \) clearly converges to the integer \(-1.\)

The sequence \( (4), {\large (4)^ \frac{1}{(4)} }, {\Large (4) ^ \frac{1}{(4)^ \frac{1}{(4)}} }, \ldots \) converges to the integer 2. (Can you prove it?)

Does there exist another value \( b \neq 1, -1, 4 \) such that the sequence \(\{a_n\}\) defined recursively by \[a_0 = b; \quad a_{n+1} = {\large b^{\frac{1}{a_n}} }\] also converges to an integer?

Note: Written out, this sequence is \( (b), {\large (b)^ \frac{1}{(b)} }, {\Large (b) ^ \frac{1}{(b)^ \frac{1}{(b)}} }, \ldots. \)

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