Calculating infinite series of roots

[ijabbott]

A "power tower" of \(n\) \(x\)'s \(\underbrace{x^{x^{x^{\dots^x}}}}_n\) is also known as the \(n\)th tetration of \(x\). The operation is called "tetration" because it is the fourth in a series of operations: addition, multiplication, exponentiation, tetration. There are other terms for the same thing. There are also several notations, but one such notation is \(^{n}x\).

Now we come on to the "infinite tetration" of \(x\), or \(^{\infty}x\), which is \(\lim\limits_{n\to\infty}{^{n}x}\). For certain real values of \(x\) such as \(x=\sqrt{2}\), this infinite tetration converges (to \(2\) in this case). In fact it only converges for real \(x\) in the interval \(\big[e^{-e}, e^\frac{1}{e}\big]\) (approx. \([0.066, 1.445]\)).

The limit of convergence can evaluated using the Lambert W function:

\[^{\infty}x = \lim_{n\to\infty}{^{n}x} = \frac{W(-\ln(x))}{-\ln(x)} \big|_{e^{-e} \le x \le e^\frac{1}{e}}\]

where the Lambert W function is defined by \(W(z{e^z})=z\), or by \(z_0 = W(z_0)e^{W(z_0)}\).

Unfortunately, the Lambert W function is not yet built-in on the HP Prime, although it is available in Xcas/Giac, so maybe later....