convergence of a series using the root test

Question

I want to find whether this series diverges or converges using the root test $$ \sum_{n=1}^\infty \arccos^n \left(\frac{1}{n^2}\right) $$ $$ \sqrt[n] {\arccos^n \left(\frac{1}{n^2}\right)} $$ $$ \arccos \sqrt[n] {\left(\frac{1}{n^2}\right)} $$ $$ \arccos 1 = 0 $$ and since its 0 the series should converge but I am not sure about the calculations there.

Answer 1

The series diverges. Indeed $$\arccos^n\left(\frac{1}{n^2}\right)\sim \left(\frac{\pi }{2}\right)^n+O\left(\frac{1}{n^2}\right);\;n\to\infty$$ The first $50$ terms sum is about $1.7\times 10^{10}$.

Answer 2

Recall that

$$\arccos(x) = \frac{\pi}{2} - \arcsin(x) =\frac{\pi}{2}-x+O(x^2) \to \frac \pi 2 $$

therefore the series diverges since $a_n \not \to 0$.