Example 8. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^{2}+1}$.
Solution. We propose to verify that the D'Alembert and Cauchy (radical) criteria are not applicable, as in this case \( q=1 \). We will apply the Cauchy integral test.

\[
\begin{aligned}
& \text { We have } a_{n}=\frac{1}{n^{2}+1} . \text { Let } f(x)=\frac{1}{x^{2}+1} . \text { Then } f(n)=a_{n} \text { and } \\
& \int_{1}^{+\infty} f(x) d x=\lim _{A \rightarrow+\infty} \int_{1}^{A} \frac{d x}{x^{2}+1}=\lim _{A \rightarrow \infty}(\operatorname{arctg} A-\operatorname{arctg} 1)=\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}
\end{aligned}
\]

The convergence of the improper integral implies the convergence of the series.

Answer. The series converges.