Boundedness of the sequence n^n^(-x)!

1. Nov 10, 2012

bins4wins

For any real $x > 0$, prove that the sequence $n^{n^{-x}}$ is bounded (and if possible, monotonically decreasing after some point). The catch is that logarithms and the exponential constant cannot be used. We must arrive at the proof using fairly "primitive tools"

If you look at the graph of the function $f(n) = n^{n^{-x}}$ you'll notice that it increases at first but then decreases asymptotically towards 1 after some point. My attempt at a solution consisted of choosing three integers $a < b < c$, then assuming that if $a^{a^{-x}} > b^{b^{-x}}$ then $b^{b^{-x}} > c^{c^{-x}}$, but I'm not sure if there is enough information in the hypothesis to prove the desired.

Last edited: Nov 10, 2012
2. Nov 10, 2012

SammyS

Staff Emeritus
Hello bins4wins. Welcome to PF !

Use double $,$​\$, for LaTeX tags

or

use double #, #​#, for inline LaTeX tags

(I used double # in the first line of the above quoted message.)

3. Nov 10, 2012

bins4wins

Thanks, I was wondering why it wasn't working...

4. Nov 10, 2012

haruspex

It looks useful to consider the ratio an+1/an, raised to the power of nx.