# How to find this limit: A=limn→∞√1+√12+√13+⋯+√1nA=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}

Question:

Show that
exists, and find the best estimate limit $A$.

It is easy to show that

and it is well known that this limit
exists.

So

But can use some math methods to find an approximation to this $A$ by hand?

and I guess maybe this is true:

By the way: we can prove $A$ is a transcendental number?

Thank you very much!

As far as numeric approximations are concerned, $\displaystyle{A\simeq\frac{(\pi+1)\ln4}{1+\ln16}}$ comes close, within an error of less than $10^{-8}$.