tiny q: superroots of real numbers x>e

[bo198214]

On the other hand, it should arrive at 3^(1/3)...

Do I actually overlook something and the sequence can indeed cross e^(1/e)?

Indeed a very interesting observation, Gottfried.

You only arrive at the expected value if it is \( <e \), i.e.
\( \lim_{k\to\infty} \text{srt}(x,k)=\sqrt[x]{x} \) only if \( 1\le x\le e \).

This is because \( 1\le {^\infty}b\le e \) for \( 1\le b\le e^{1/e} \), where \( x={^\infty}b \) and \( b=\sqrt[x]{x} \).

For \( x>e \), for example \( x=3 \), is always \( \text{srt}(x,k) > e^{1/e} \) for each \( k \). Suppose otherwise \( \text{srt}(x,k)\le e^{1/e}=:y \) then would \( x\le {^k}y \), for \( y\le e^{1/e} \) while \( {^k}y\le e \).