# Power Series

## Trigonometry # Power Series

## Definition

A series, terms of which are power functions of variable $$x,$$ is called the power series:

$\sum\limits_{n = 1}^\infty {{a_n}{x^n}} = {a_0} + {a_1}x + {a_2}{x^2} + \ldots + {a_n}{x^n} + \ldots$

A series in $$\left( {x - {x_0}} \right)$$ is also often considered. This power series is written as

$\sum\limits_{n = 1}^\infty {{a_n}{{\left( {x - {x_0}} \right)}^n}} = {a_0} + {a_1}\left( {x - {x_0}} \right) + {a_2}{\left( {x - {x_0}} \right)^2} + \ldots + {a_n}{\left( {x - {x_0}} \right)^n} + \ldots ,$

where $${x_0}$$ is a real number.

## The Interval and Radius of Convergence

Consider the function

$f\left( x \right) = \sum\limits_{n = 1}^\infty {{a_n}{{\left( {x - {x_0}} \right)}^n}}.$

The domain of this function is the set of those values of $$x$$ for which the series is convergent. The domain of such function is called the interval of convergence.

If the interval is $$\left( {{x_0} - R,{x_0} + R} \right)$$ for some $$R \gt 0,$$ (together with one or both of the endpoints), the $$R$$ is called the radius of convergence. Convergence of the series at the endpoints is determined separately.

Using the root test, the radius of convergence is given by the formula

$R = \lim\limits_{n \to \infty } \frac{1}{{\sqrt[n]{{{a_n}}}}},$

but a fast way to compute it is based on the ratio test:

$R = \lim\limits_{n \to \infty } \left| {\frac{{{a_n}}}{{{a_{n + 1}}}}} \right|.$

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Find the radius of convergence and interval of convergence of the power series $\sum\limits_{n = 0}^\infty {\frac{{{{\left( {x + 3} \right)}^n}}}{{n!}}}.$

### Example 2

Determine the radius of convergence and interval of convergence of the power series $\sum\limits_{n = 0}^\infty {n{x^n}}.$

### Example 1.

Find the radius of convergence and interval of convergence of the power series $\sum\limits_{n = 0}^\infty {\frac{{{{\left( {x + 3} \right)}^n}}}{{n!}}}.$

Solution.

We make the substitution: $$u = x + 3.$$ The series becomes $$\sum\limits_{n = 0}^\infty {\frac{{{u^n}}}{{n!}}}.$$ Calculate the radius of convergence:

$R = \lim\limits_{n \to \infty } \left| {\frac{{{a_n}}}{{{a_{n + 1}}}}} \right| = \lim\limits_{n \to \infty } \frac{{\frac{1}{{n!}}}}{{\frac{1}{{\left( {n + 1} \right)!}}}} = \lim\limits_{n \to \infty } \frac{{\left( {n + 1} \right)!}}{{n!}} = \lim\limits_{n \to \infty } \left( {n + 1} \right) = \infty .$

Then the interval of convergence is $$\left( { - \infty ,\infty } \right).$$

### Example 2.

Determine the radius of convergence and interval of convergence of the power series $\sum\limits_{n = 0}^\infty {n{x^n}}.$

Solution.

Calculate the radius of convergence:

$R = \lim\limits_{n \to \infty } \left| {\frac{{{a_n}}}{{{a_{n + 1}}}}} \right| = \lim\limits_{n \to \infty } \frac{n}{{n + 1}} = \lim\limits_{n \to \infty } \frac{1}{{1 + \frac{1}{n}}} = 1.$

Consider convergence at the endpoints.

If $$x = -1,$$ we have the divergent series $$\sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}n}.$$

If $$x = 1,$$ the series $$\sum\limits_{n = 0}^\infty n$$ is also divergent.

Therefore, the initial series $$\sum\limits_{n = 0}^\infty {n{x^n}}$$ converges in the open interval $$\left( { -1, 1} \right).$$