you wrote:

“As far as I understand, Chris did not mean to apply this formula iteratively.”

Yes and no. Do you remember my example when I computed sqrt(66) with two
iterations? With a “good” initial guess you can use the formula to approximate a root
in one step, but with a “poor” initial guess you need more iterations, of course.

However, I would like to repeat John’s observation:

Assume we want to compute the nth root(1000), the initial guess is always=1, desired accuracy: EXCEL’s 14 digits.

Now let n increase, say, 2, 3, 4, 5, 10, 100, 1000. Then you will see that the number of iterations needed to reach the desired accuracy does not increase – contrary to Newton’s method. To compute the 1000th root(1000) Newton’s algorithm needs 692 iterations, the new algorithm needs (always) 7!

So it'seems that the number of iterations does not depend on n.
I’ve got absolutely no explanation for this property, but maybe it’s possible to prove
it mathematically, as John wrote.

By the way, the algorithm also converges if n is negative.

Kind regards

Chris