Bounding $|n^2(1-(\cos\frac{1}{n})^2)|$

Question

I'm working on a a problem that involves me needing to give an upper bound for the following expression:

$|n^2(1-(\cos\frac{1}{n})^2)|$

My attempts at bounding it:

Expanding the expression:

$|n^2-n^2\cos^2\frac{1}{n}|\leq n^2-n^2(1)=0$, so I'm pretty sure I'm not doing it correctly...

Anyone have any suggestions? Thanks.

Answer 1

You probably want to do a second order Taylor approximation to $\cos$:

$$\cos\left(\frac{1}{n}\right) \approx 1 - \frac{1}{2n^2}.$$