# Upper bounds for $f_k(n)$ (functions in the fast growing hierachy) [closed]

Mostly, for the functions in the fast growing hierachy, LOWER bounds are given like

$f_k(n) > 2 \uparrow^{k-1} n$

but what abour (reasonable tight) UPPER bounds ?

What are the best known UPPER bounds for $f_k(n)$ ?

The functions in the fast growing hierachy are defined as follows

$f_0(n) = n+1$

$f_{k+1} = f_k^n(n) = f_k(f_k(...n)...)$

where $f_k$ appears n times.

So $$f_1(n) = 2n$$ $$f_2(n) = n2^n$$ $$f_3(n) \ge 2 \uparrow 2 \uparrow ... \uparrow 2 \uparrow n$$

with n 2's

and so on.

## closed as unclear what you're asking by Stefan Kohl, Ryan Budney, Boris Bukh, user9072, S. Carnahan♦Apr 30 '14 at 3:10

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• What sort of notation do you want to use to express upper bounds when $k$ is large? At some point, you will need a recursive formula. – S. Carnahan Apr 30 '14 at 3:10