Conjecture
\( \lim_{n\to\infty} f(n) = e^{1/e} \) where \( f(n) = x \) such that \( {}^{n}x = n \)
Discussion
To evaluate f at real numbers, an extension of tetration is required, but to evaluate f at positive integers, only real-valued exponentiation is needed. Thus the sequence given by the solutions of the equations
The conjecture is proven to be true. Search the forum for "TPID 6".
\( \lim_{n\to\infty} f(n) = e^{1/e} \) where \( f(n) = x \) such that \( {}^{n}x = n \)
Discussion
To evaluate f at real numbers, an extension of tetration is required, but to evaluate f at positive integers, only real-valued exponentiation is needed. Thus the sequence given by the solutions of the equations
- \( x = 1 \)
- \( x^x = 2 \)
- \( x^{x^x} = 3 \)
- \( x^{x^{x^x}} = 4 \)
The conjecture is proven to be true. Search the forum for "TPID 6".