# Closed Form Solution to Exponential Recursion

Is there a closed form solution to the function $f_n=2^{f_{n-1}}$ where $f_0=2$ ?

For instance, the first few values of the function are 2, 4, 16, 65536.

Depends what you mean by closed form.

If you mean to ask, does this have an expression in terms of elementary functions such as exponentiation, trigonometry, and polynomials, then the answer is NO.

How do I know that?

Because this function you've given is a famous NON elementary function, known as tetration. Read about it here!

https://en.wikipedia.org/wiki/Tetration

But an actual proof, arises from the fact that elementary functions have individual constitutents (that can be composed and added and multiplied) that grow at most as fast as $a^x$ for some base $a$.

So a finite elementary expression will have at most finitely many

$$a^{a^{a^{... ^{x}}}}$$ But this function you've given grows at a rate that is UNBOUNDED in the number of nested exponentials. So any finite elementary function will be eclipsed by it.

• Thanks, this answered my question perfectly. :) – Reubend Apr 14 '16 at 17:26