Convergence or divergence of (i) $\sum_{n=1}^{\infty} \cos \frac 1n$,(ii)$\sum_{n=1}^{\infty} \cos \frac {1}{n^2}$

Question

In case of (ii),

As $n \rightarrow \infty$, $\cos \frac {1}{n^2} \rightarrow 1 \neq 0$. Hence this series diverges.

In case of (i),

$\cos \frac 1n = 1 - 2\sin^2 \frac {1}{2n}$. But since $\sum \sin \frac {1}{k}$ diverges, Hence $\sum_{n=1}^{\infty} 1 - 2\sin^2 \frac{1}{2n}$ also. Hence $\sum \cos \frac 1n$ diverges. Am I correct here? If not,then what should I do?

Answer 1

You can use the same approach for both cases. In both cases, the $nth$ term approaches $1$ as $n$ approaches $\infty$. Any series whose nth term approaches a nonzero value diverges.

Answer 2

Both diverges , look where nth term goes as $n\to \infty$