Example 3. Investigate the convergence of the series $\sum_{n=2}^{\infty} \frac{n-1}{3 n+20}$.
Solution. Let's check the necessary condition for convergence (Theorem 1). We have \(\lim _{n \rightarrow+\infty} u_{n}=\lim _{n \rightarrow \infty} \frac{n-1}{3 n+20}=\lim _{n \rightarrow \infty} \frac{n\left(1-\frac{1}{20}\right)}{n\left(3+\frac{20}{n}\right)}=\frac{1}{3} \neq 0\).

The series diverges. Since its terms are positive, the series diverges to \(+\infty\).
Answer. The series diverges.

Example 4. Investigate the convergence of the series \(\sum_{n=1}^{\infty} \cos \pi n\).

Solution. We have \(\lim _{n \rightarrow \infty} \cos \pi n=\lim _{n \rightarrow \infty}(-1)^{n} \neq 0\) (the limit does not exist). According to Theorem 1, the series diverges. Additionally, we have: \(S_{1}=\cos \pi=-1, \quad S_{2}=\cos \pi+\cos 2 \pi=-1+1=0\), \(S_{3}=\cos \pi+\cos 2 \pi+\cos 3 \pi=-1+1-1=-1, \ldots, \lim _{n \rightarrow+\infty} S_{n}\) does not exist at all.

Answer. The series diverges.