Example 11. Investigate the convergence of the series $\sum_{n=1}^{\infty} \ln \left(1+\frac{5}{n^{2}}\right)$.
Solution. We take the convergent series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ as a comparison series. Since $\lim _{n \rightarrow \infty} \frac{\ln \left(1+\frac{5}{n^{2}}\right)}{\frac{1}{n^{2}}}=5$ (a consequence of the second remarkable limit), the series under investigation converges.

Answer. The series converges.

To increase the number of standard, or template, series (convergent or divergent), let's recall the known equivalent infinitesimal sequences:

$$
\sin \frac{1}{n} \approx \operatorname{tg} \frac{1}{n} \approx \arcsin \frac{1}{n} \approx \operatorname{arctg} \frac{1}{n} \approx \ln \left(1+\frac{1}{n}\right) \approx e^{1 / n}-1 \approx \frac{1}{n}
$$

as $n \rightarrow \infty$.

## Exercises

Find the sums of the series.

1. $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$
2. $\sum_{n=1}^{\infty} \frac{1}{4 n^{2}-9}$.
3. $\sum_{n=0}^{\infty} q^{n}(|q|<1)$
4. $\sum_{n=1}^{\infty} \sin \frac{\pi n}{2}$.
5. $\sum_{n=1}^{\infty} n \cdot \sin \frac{1}{\sqrt{n}}$.
6. $\sum_{n=1}^{\infty} n \cdot \ln \left(1+\frac{1}{n}\right)$.
7. $\sum_{n=1}^{\infty}\left(\frac{3 n-1}{2 n+7}\right)^{n / 2}$.
8. $\sum_{n=1}^{\infty} \frac{1 \cdot 4 \cdot 7 \cdot \ldots \cdot(3 n-2)}{7 \cdot 9 \cdot 11 \cdot \ldots \cdot(2 n+5)}$.

8 K.N. Lungu, E.V. Makarov

Prove the convergence of the series.
9. $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot(2 n-1)}{2 \cdot 5 \cdot 8 \cdot \ldots \cdot(3 n-1)}$
10. $\sum_{n=1}^{\infty} \frac{n!(2 n-1)!}{(3 n)!}$
11. $\sum_{n=2}^{\infty} \frac{1}{n \ln ^{3} n}$
12. $\sum_{n=1}^{\infty} \frac{2^{n} \cdot n!}{n^{n}}$.
13. $\sum_{n=1}^{\infty} \frac{1}{n \sqrt{n+1}}$.
14. $\sum_{n=1}^{\infty} \frac{3^{n-2} \sqrt{n^{2}+7}}{(n+1)!}$.
15. $\sum_{n=1}^{\infty} \frac{2^{n}-1}{3^{n}+1}$
16. $\sum_{n=1}^{\infty} \frac{n^{2}+2}{n^{3} \ln ^{2} n}$.
17. $\sum_{n=1}^{\infty} \frac{2 n^{2}+5 n-3}{n^{5}-3 n^{2}+2}$

Answers

1. 2. 2. $\frac{1}{18} \cdot$ 3. $\frac{1}{1-q} \cdot$ 4. $\frac{1}{2} \cdot$ 5. $\frac{1}{6} \cdot$ 6. 1. 7. 2.5. 8. $\frac{1}{8}$.

## § 2. Convergence of Alternating Series

$1^{\circ}$. A numerical series $\sum_{n=1}^{\infty} v_{n}$ is called alternating if among its terms there are both positive and negative ones.

Theorem 10. If the series $\sum_{n=1}^{\infty}\left|v_{n}\right|$ of absolute values of the terms of the series $\sum_{n=1}^{\infty} v_{n}$ converges, then the series $\sum_{n=1}^{\infty} v_{n}$ also converges.

$2^{\circ}$. If an alternating series $\sum_{n=1}^{\infty} v_{n}$ converges, then the positive series $\sum_{n=1}^{\infty}\left|v_{n}\right|$ may diverge. This justifies the following definition:

If, along with the convergent alternating series $\sum_{n=1}^{\infty} v_{n}$, the series $\sum_{n=1}^{\infty}\left|v_{n}\right|$ also converges, then the series $\sum_{n=1}^{\infty} v_{n}$ is called absolutely convergent.

If the series $\sum_{n=1}^{\infty} v_{n}$ converges, but the series $\sum_{n=1}^{\infty}\left|v_{n}\right|$ diverges, then the series $\sum_{n=1}^{\infty} v_{n}$ is called conditionally convergent, or non-absolutely convergent.

$3^{\circ}$. A special case of an alternating series is the alternating series $\left(u_{n}>0\right)$:

$$
\begin{gathered}
u_{1}-u_{2}+u_{3}-u_{4}+\ldots+u_{2 n-1}-u_{2 n}+u_{2 n+1}-u_{2 n+2}+\ldots= \\
=\sum_{n=1}^{\infty}(-1)^{n-1} \cdot u_{n}
\end{gathered}
$$

Theorem 11 (Leibniz). Suppose the alternating series $\sum_{n=1}^{\infty}(-1)^{n-1} \cdot u_{n}, u_{n}>0$, satisfies two conditions:

1) $u_{1}>u_{2}>u_{3}>\ldots>u_{n}>\ldots\left(\left\{u_{n}\right\}_{n=1}^{\infty}\right.$ is monotonically decreasing);
2) $\lim _{n \rightarrow \infty} u_{n}=0$

Then the series converges and its sum $S0$, it is sufficient to compute $S_{n}$, where $n=1$ is the smallest number for which $u_{n+1}<\varepsilon$.

## Examples with Solutions