Example 2. Find the domain of convergence of the series

$$
\sum_{n=1}^{\infty} \frac{4^{n}}{n^{3}\left(x^{2}-4 x+7\right)^{n}}
$$
Solution.

1. To apply the D'Alembert's criterion, we find $\varrho(x)$ using the formula

$$
\varrho(x)=\lim _{n \rightarrow \infty} \frac{\left|f_{n+1}\right|}{\left|f_{n}\right|}=\lim _{n \rightarrow \infty} \frac{\frac{4^{n+1}}{(n+1)^{3}\left|x^{2}-4 x+7\right|^{n+1}}}{\frac{4^{n}}{n^{3}\left|x^{2}-4 x+7\right|^{n}}}=\frac{4}{\left|x^{2}-4 x+7\right|}
$$

2. Since by D'Alembert's criterion the series converges when $\varrho<1$ and diverges when $\varrho>1$, we find the interval of convergence by solving the inequality $\varrho(x)<1$ (i.e., $\left|x^{2}-4 x+7\right|>4$). The solution to this inequality is $x \in (-\infty, 1) \cup (3, +\infty)$. At the boundary points, when $x=1$, the series

$$
\sum_{n=1}^{\infty} \frac{1}{n^{3}}
$$

converges; when $x=3$, the series

$$
\sum_{n=1}^{\infty} \frac{1}{n^{3}}
$$

also converges.

Answer. The interval of convergence of the series is $(-\infty, 1] \cup [3, +\infty)$.

Conditions of the Problem. Find the intervals of convergence of the functional series.

1. $\sum_{n=1}^{\infty} \frac{n}{(\sqrt[3]{n}+1)^{x+3}}$.
2. $\sum_{n=1}^{\infty} \frac{2^{n}}{n\left(x^{2}-6 x+10\right)^{n}}$.
3. $\sum_{n=1}^{\infty} \frac{n^{4}}{n^{x^{2}+4 x}+3}$.
4. $\sum_{n=1}^{\infty} \frac{3^{n}}{n\left(x^{2}-5 x+9\right)^{n}}$.
5. $\sum_{n=1}^{\infty} \frac{n}{n^{x^{2}-2}+1}$.
6. $\sum_{n=1}^{\infty} \frac{4^{n}}{n^{2}\left(x^{2}+3\right)^{n}}$.
7. $\sum_{n=1}^{\infty} \frac{n^{2}}{(n+\sqrt{n})^{x}}$.
8. $\sum_{n=1}^{\infty} \frac{5^{n}}{n\left(x^{2}-4 x+8\right)^{n}}$.
9. $\sum_{n=1}^{\infty} \frac{n^{2}+1}{n^{3+3 x-x^{2}}}$.
10. $\sum_{n=1}^{\infty} \frac{6^{n}}{n^{2}\left(x^{2}-2 x+6\right)^{n}}$.

Answers. 1. $(3,+\infty)$. 2. $(-\infty, 2) \cup (4,+\infty)$. 3. $(-\infty,-5) \cup (1,+\infty)$. 4. $(-\infty, 2) \cup (3,+\infty)$. 5. $(-\infty,-2) \cup (2,+\infty)$. 6. $(-\infty,-1] \cup [1,+\infty)$. 7. $(3,+\infty)$. 8. $(-\infty, 1) \cup (3,+\infty)$. 9. $(0,3)$. 10. $(-\infty, 0] \cup [2,+\infty)$.

### 10.10. Interval of Convergence of a Power Series

Problem Statement. Find the interval of convergence of the power series

$$
\sum_{n=1}^{\infty} c_{n}\left(x-x_{0}\right)^{n}
$$

Plan of Solution.

1. If the limits

$$
\lim _{n \rightarrow \infty} \frac{\left|c_{n}\right|}{\left|c_{n+1}\right|} \text { or } \frac{1}{\lim _{n \rightarrow \infty} \sqrt[n]{\left|c_{n}\right|}}
$$

exist, then the radius of convergence of the power series can be found using D'Alembert's formula

$$
R=\lim _{n \rightarrow \infty} \frac{\left|c_{n}\right|}{\left|c_{n+1}\right|}
$$

or Cauchy's formula

$$
R=\frac{1}{\lim _{n \rightarrow \infty} \sqrt[n]{\left|c_{n}\right|}}
$$

Then the interval of convergence of the series is $-R < x - x_0 < R$. Therefore, we find the interval of convergence by solving the inequality $\varrho(x) < 1$.

2. Investigate the behavior of the power series at the boundary points of the interval of convergence $x = x_0 \pm R$.

Write the answer.