Nested Fractional Exponent Discovered

Calculus Level 3

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

The sequence (4),(4)1(4),(4)1(4)1(4), (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 b1,1,4 b \neq 1, -1, 4 such that the sequence {an}\{a_n\} defined recursively by a0=b;an+1=b1ana_0 = b; \quad a_{n+1} = {\large b^{\frac{1}{a_n}} } also converges to an integer?

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

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