Convergence of a Series using the p-series Approach

Question

Determine if $\sum\limits_{n = 1}^{\infty} \cos\left(\frac{1}{n^2}\right)$ converges using the $p$-series approach.

Answer: Is this a simple as saying that $$\sum\limits_{n = 1}^{\infty} \cos\left(\frac{1}{n^2}\right) = \cos\left(\sum\limits_{n = 1}^{\infty} \frac{1}{n^2}\right)$$

and since $\sum\limits_{n = 1}^{\infty} \frac{1}{n^2}$ is a convergent $p$-series, then its sum is finite, and the cosine of a finite number is itself a finite number, hence the given series is convergent?

Answer 1

Since $\lim\limits_{n\to\infty} \cos\left(\frac{1}{n^2}\right) = \cos\left(\lim\limits_{n\to\infty}\frac{1}{n^2}\right) = \cos(0) = 1 \neq 0$, then the given series $\sum\limits_{n = 1}^{\infty} \cos\left(\frac{1}{n^2}\right)$ diverges.