I was trying to find the answer to $$x\sqrt{x\sqrt[3]{x\sqrt[4]{x...\sqrt[n]{x\dots}}}}$$ I eventually got $x^e$ but I can't find a notation to write that in so wolfram alpha understands it and I tried graphing it to see if it approaches $x^e$ and it doesn't.
So I think I am incorrect, but what I did is I put all the $x$'s in the sqrt \begin{align} &= \lim_{n \to \infty} ((x * (x^{(n)(n-1)}) * x^{n(n-1)(n-2)} ... x^0))^{1/n!}\\ &= x^{\frac{1}{n!}\sum_{k=0}^{\infty}\frac{n!}{\left(n-k\right)!}}\\ &= x^{\sum_{k=0}^{\infty}\frac{1}{k!}}\\ &= x^e \end{align}
I want to know how I can verify the result, or if I am correct.