Tetration towers

Calculus Level 3

Consider the infinitely nested exponential equation

$\large x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} = N.$

One might naively say, "Easy, just substitute in," and

$x^N = N, \ \text{ so }\ x = \sqrt[N]{N}.$

However, this doesn't converge for all $N$ . What is the highest $N$ for which it does?

Give your answer to 3 decimal places.

The answer is 2.718.

1 solution

Geoff Pilling
May 19, 2016

Yes, the highest number, is, you guessed it... Our favorite constant, $e \approx \boxed{2.718}$ , courtesy of our very own... Euler!

The proof is left as an exercise to the reader! :)

We discussed this problem here , with various solutions.

Otto Bretscher - 5 years ago

Ooops, you are right... It is pretty much the same question... :-/ Sorry for the repeat...

Geoff Pilling - 5 years ago

It's a fun problem, and it's good that people see it from time to time. I wasn't the first to ask this question either; that was probably my country man Leonhard Euler ;)

I just wanted to help you out and point out that solutions are available ;)

Otto Bretscher - 5 years ago

@Otto Bretscher – I hadn't seen it before, but was curious when I discovered that 2 converged, but 3 didn't. So, I had to try it out for myself last night with a spreadsheet, and sure enough it was e where it starts to diverge! :) But, what I was quite surprised about was that as you approach e it takes longer and longer to converge. It takes a very long time for example for 2.718.

Geoff Pilling - 5 years ago

@Geoff Pilling – Yes indeed. You get a sense of why this happens when you look at the cobwebs.

It is also interesting to see what happens when you choose N between 0 and 1.

Otto Bretscher - 5 years ago

Tetration towers

The answer is 2.718.

1 solution

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