Radius of Convergence

When you are practicing throwing a ball at a target, you start by standing in one spot until you can hit the target multiple times. Then you start to wonder how far you can move from your original spot and still hit the target. Maybe you can move a foot from your starting point and still hit the target, but any further and you will miss. That distance from the starting point where things still work out is like the radius of convergence of a series, and the actual space you can move around in and still hit the target is like the interval of convergence.

A popular types of series are the power series, which are series of the form \(\sum_{n}^\infty c_nx^n\), whose convergence will depend on the value of \(x\). In this article, you will see how to find the values of \(x\), for which the power series converges. In other words, you will learn how to calculate its radius of convergence and its interval of convergence.

Definition of Radius of Convergence

As mentioned above, the convergence of the power series depends on the values of \(x\).

For a power series

\[\sum\limits_{n=1}^\infty c_n(x-x_0)^n ,\]

the relationship between the radius of convergence with the interval of convergence is shown in the table below.

Table 1. Relationship between radius of convergence and interval of convergence.

Radius of Convergence of Power Series

A power series centered at \(x_0\) (or a power series about \(x_0\)) is a series of the form

\[\sum_{n=0}^\infty c_n(x-x_0)^n=c_0+c_1(x-x_0)+c_2(x-x_0)^2+\dots,\]

where \(x\) is a variable, and \(x_0\) and \(c_n\) are real numbers. The \(c_n\)'s are called the coefficients of the series.

For each value of \(x\), the series can converge or diverge. The radius of convergence of a series is what will tell you for which values of \(x\) it converges.

Ratio Test and Radius of Convergence

To calculate the radius of convergence of a power series, you can use the Ratio Test (or sometimes the Root Test).

Let's see an example of how to use the Ratio Test to obtain the radius of convergence.

Radius of Convergence of Geometric Series

A special case of power series is the geometric series given by

\[\sum\limits_{n=0}^\infty ax^n,\]

where \(a\) is a constant.

You can calculate its radius of convergence using the Ratio Test just like for other power series. In this case, the terms of the series are given by \(a_n=ax^n\), so

\[\begin{align} \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| &=\lim_{n\to\infty}\left|\frac{ax^{n+1}}{ax^n}\right| \\ &=|x|. \end{align}\]

The Ratio Test says that it converges if \(|x|<1\), and therefore the radius of convergence is \(R=1\).

To calculate the radius of convergence of the function \(\sin x\), remember that you can rewrite the function using its Taylor or Maclaurin series. The Maclaurin series of the function \(\sin x\) is given by

\[\begin{align} \sin x&=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1} \\ &=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots \end{align}\]

\[\begin{align}\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| & =\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}}{(2(n+1)+1)!}x^{2(n+1)+1}}{\frac{(-1)^n}{(2n+1)!}x^{2n+1}}\right| \\ &=\lim_{n\to\infty}\frac{(2n+1)!}{(2n+3)!}|x|^2 \\ &= \lim_{n\to\infty}\frac{1}{(2n+3)(2n+2)}|x|^2 \\ &=0. \end{align}\]

Since the limit is zero independently of \(x\), it means that the series converges for any value of \(x\), and therefore the series of \(\sin x\) has radius of convergence \(R=\infty\).

Examples of Radius of Convergence

Here are some examples.

One more example.