Infinite Exponentials

Calculus Level pending

The field of continuing fractions and infinite compounded functions is a very interesting one. To get started, solve this equation where n is an integer:

$x^{x^{x^{...}}} = n$

What is $x$ ?

n^{2n}

n^{1/n}

n^{n}

n^{n^2}

1 solution

Imran Qureshi
Mar 23, 2016

Did you get it? Thinking about the problem directly, it seems quite difficult as you have to think about more and more exponentials in the equation. To make this easier, lets pick a more definite n to start, say n=2:

$x^{x^{x^{...}}} = 2$

Now since the x's go on to infinity, we can do a neat trick. That is replace the top portion as if it was the whole equation:

$x^{2} = 2$

Seems like we did something strange, but this is totally valid. And now the problem is way easier. The solution is simply:

$x = 2^{1/2}$

Let's think about the general case. Just as we did in the n=2 example, simply perform the same operation and you will get:

$x^n = n \implies x = n^{1/n}$

Infinite Exponentials

1 solution

0 pending reports