For any number xx, we can express the difference between xx and xx raised to some power nn as:


To find which number xx most exceeds xnx^n, we can find the maximum point of the above function.

First the derivative:


And then the maximum (assuming that the power, nn, is greater than 1):





We can relabel this final equation as a function in xx and yy which will accept a power and yield the number which most exceeds itself raised to that power. In other words, a function which yields all the maximums of our original function (y=xxny=x-x^n).


This function can be shown to converge to 11:

limx (1x)1/(x1)=limx (1x)0=limx 1=1\lim_{x \to \infty}~(\frac{1}{x})^{1/(x-1)} = \lim_{x \to \infty}~(\frac{1}{x})^{0}=\lim_{x \to \infty}~1=\boxed{1}

This is equivalent to saying that the number nn which most exceeds nn^{\infty} is 11 (11 most exceeds 11^{\infty}). Remember that (because it gets weirder!).

Now the function describing the difference between these "greatest" numbers and their respective powers is:








This can also be shown to converge to 11:

limx (11x)(1x)1/(x1)=limx (10)(1x)0=limx 1=1\lim_{x \to \infty}~(1-\frac{1}{x})(\frac{1}{x})^{1/(x-1)} = \lim_{x \to \infty}~(1-0)(\frac{1}{x})^{0}=\lim_{x \to \infty}~1=\boxed{1}

This is equivalent to saying that the difference between the number nn which most exceeds nn^{\infty} (11) and nn^{\infty} (11^{\infty}) is 11.

This is understandably counter-intuitive, but taken in this context, it seems provable!

To finish, here is the grand consequence of this discussion:


Which means:



Note by David Stiff
2 years ago

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I think it depends on the “1”. Think of the context of e.

Ruilin Wang - 1 year, 10 months ago

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I know, it really can't be taken literally. Reading this again months after I wrote it, I actually had a hard time following along! I think I had been inspired by something in my math textbook at the time.

David Stiff - 1 year, 10 months ago

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Ruilin Wang - 1 year, 10 months ago

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Actually I discovered an "error" in this absurdity which makes 1=21^{\infty}=2. :)

David Stiff - 1 year, 10 months ago

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