TITLE: Find $\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{a}}{a}}}{a}}}{\begin{array}{c} a\\\vdots \end{array}}}$
QUESTION [5 upvotes]: Assuming $m\in \Bbb N\setminus\{0,1\}$ and $a\in \Bbb R_+\setminus\{0\}$, find 
  $$\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{a}}{a}}}{a}}}{\begin{array}{c}
a\\\vdots
\end{array}}}$$

In a book I'm studying the author gives $$a^{\frac{m}{m+1}}$$ as the answer. But I'm a little uncomfortable with the development presented. In an example, in which $a=128$ and $m=6$, the numeric result given is 64 (by applying the formula).
But, trying to compute a numerical evaluation of the sequence, it appears to converge to something less then 0.38. 
Confirmation of the result or a correct answer to the problem will be appreciated. Sorry if this is a duplicate.

REPLY [2 votes]: For $m>1$, $a>0$ and $x_0>0$, the recurrence $x_{n+1}=\frac a{\sqrt [m]{x_n}} $
has limit $\ell $ satisfying
$\ell=\frac {\sqrt [m]{\ell}}a $
from which $\ell=a^{\frac m {1+m} }$.
Hence for $a=128$ and $m=6$ you get the limit $\ell=64$.