Example 5.9. Based on the D'Alembert's criterion, investigate the convergence of the series

$$
3+\frac{3^{2}}{2^{2}}+\frac{3^{3}}{3^{3}}+\frac{3^{4}}{4^{4}}+\ldots+\frac{3^{n}}{n^{n}}+\ldots
$$
Solution. Knowing the $n$-th term of the series $a_{n}=\frac{3^{n}}{n^{n}}$, we write the $(n+1)$-th term: $a_{n+1}=\frac{3^{n+1}}{(n+1)^{n+1}} \cdot$ From this,

\[
\begin{gathered}
\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\lim _{n \rightarrow \infty} \frac{3^{n+1}}{(n+1)^{n+1}}: \frac{3^{n}}{n^{n}}=\lim _{n \rightarrow \infty} \frac{3^{n+1} \cdot n^{n}}{3^{n}(n+1)^{n+1}}=\lim _{n \rightarrow \infty} \frac{3}{n+1} \cdot \frac{n}{n+1}{ }^{n}= \\
=3 \lim _{n \rightarrow \infty} 1-\frac{n}{n+1}{ }^{n} \cdot \frac{1}{n+1}=3 \cdot e^{-1} \cdot 0=0
\end{gathered}
\]

Since $l=0<1$, the series converges.

## 5.3. Leibniz's Convergence Test

An alternating series is a series of the form

\[
a_{1}-a_{2}+a_{3}-\ldots+(-1)^{n-1} a_{n}+\ldots
\]

where $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ are positive numbers.

For alternating series, the following convergence test applies.

Leibniz's Test. The series (1) converges if its terms decrease monotonically in absolute value and the general term tends to zero as $n \rightarrow \infty$.

The application of convergent series to approximate calculations is based on replacing the sum of the series with the sum of its first few terms. The error introduced by this is easily estimated for an alternating series that satisfies Leibniz's test - the error is less than the absolute value of the first of the discarded terms of the series.