Calculating infinite series of roots

[ijabbott]

(08-30-2019 11:57 PM)ijabbott Wrote:  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 \(z_0 = W(z_0)e^{W(z_0)}\).

I've just noticed that this evaluates to \(\frac{0}{0}\) for \(x=1\). Hmm... more work needed? Also when \(1 \lt x \le e^\frac{1}{e}\), then letting \(z=-\ln(x)\), \(-\frac{1}{e} \le z \lt 0\), and there are two branches of the \(W\) function in this interval. I guess it uses the upper branch?