Does $A_n= \sqrt{1^2+\sqrt{2^2+\sqrt{...+\sqrt{n^2}}}}$ converge? Can we find the value of $\lim\limits_{n\to \infty} A_n$, does it converge?
$$A_n= \sqrt{1^2+\sqrt{2^2+\sqrt{3^2+\sqrt{4^2+\sqrt{...+\sqrt{n^2}}}}}} $$
I tried to calculate $A_1,A_2,\cdots A_{10}$, but my computer cannot do further terms.

See here for references to a criterion.
$$\limsup_{n\to\infty} \frac{\log n^2}{2^n} = 2\limsup_{n\to\infty} \frac{\log n}{2^n} = 0$$
So the convergence is positive. The value is another story though.
