Limit of $n^2\sqrt{1-\cos(1/n)+\sqrt{1-\cos(1/n)+\ldots}}$ when $n \to \infty$ Compute the limit:
$$\lim_{n \to \infty} n^2\sqrt{1-\cos(1/n)+\sqrt{1-\cos(1/n)+\sqrt{1-\cos(1/n)+\ldots}}}$$

Hints:


*

*For every $a\gt0$, $b=\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}$ is such that $b^2+a=b$ and $b\gt0$, thus $b=\frac12+\frac12\sqrt{1+4a}$.

*When $n\to\infty$, $1-\cos(1/n)\to0$.

*When $a\to0$, $\frac12+\frac12\sqrt{1+4a}\to1$.

*Hence the limit you are after is $\lim\limits_{n\to\infty}n^2\cdot1=+\infty$.

