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

[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.

 

[SammyS]

For any real $x > 0$, prove that the sequence $\large n^{n^{-x}}$ 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.

Hello bins4wins. Welcome to PF !

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[bins4wins]

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

 

[haruspex]

It looks useful to consider the ratio a_n+1/a_n, raised to the power of n^x.