More Fun With Power Towers

Calculus Level 2

\[\huge a^{a^{a^{\cdot^{\cdot^\cdot}}}}=b\]

Find the largest real number \(b\) such that the above equation has a positive real solution \(a\). Approximate \(b\) to four significant digits.

If you come to the conclusion that no such value of \(b\) exists, enter 666.

Clarification: The value of the infinite power tower \(\large a^{a^{a^{\cdot^{\cdot^\cdot}}}}\) is defined as the limit of the sequence \(x_0=a, x_{n+1}=a^{x_n}\).


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