Example 4.17. Investigate the convergence of the Dirichlet series $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$.
Solution. Consider the function $f(x)=\frac{1}{x^{p}}$. This function is positive and monotonic for $x \geqslant 1$, and moreover, $\forall n \in N$ the equality $f(n)=\frac{1}{n^{p}}=a_{n}$ holds. Therefore, the question of the convergence of this series reduces, by Theorem 4.14, to the investigation of the improper integral $\int_{n=1}^{\infty} \frac{d x}{x^{p}}, a \geqslant 1$:

$$
\int_{a}^{+\infty} \frac{d x}{x^{p}}=\lim _{A \rightarrow+\infty} \int_{a}^{A} \frac{d x}{x^{p}}=\left.\lim _{A \rightarrow+\infty} \frac{x^{1-p}}{1-p}\right|_{a} ^{A}=\lim _{A \rightarrow+\infty} \frac{A^{1-p}}{1-p}-\frac{a^{1-p}}{1-p}
$$

The last calculation shows that the improper integral converges for $p>1$ and diverges for $p \leqslant 1$.