# Finding the value of √1+2√2+3√3+4√4+5√5+…\sqrt{1+2\sqrt{2+3\sqrt{3+4\sqrt{4+5\sqrt{5+\dots}}}}}

Is it possible to find the value of

Does it help if I set it equal to $x$? Or I mean what can I possibly do?

I don’t see it’s going anywhere. Help appreciated!

This is meant to follow up on Ethan’s comment about using Herschfeld’s theorem to prove that the expression converges.

Theorem (Herschfeld, 1935). The sequence

converges if and only if

The American Mathematical Monthly, Vol. 42, No. 7 (Aug-Sep 1935), 419-429.

In our case we have

and so on, so that

We then have

as $n \to \infty$, where the infinite product converges because $k^{1/2^{k-1}} = 1 + O(\log k/2^k)$ as $k \to \infty$. Therefore $u_n$ converges by Herschfeld's theorem.