The Radius of Convergence


Consider the next power series.

$$\sum_{n=0}^∞ a_nz^n・・・(1)$$

We take the absolute value of \(a_n\) and \(z\) which make up this power series.

$$\sum_{n=0}^∞ |a_n||z|^n・・・(2)$$

This power series (2) is called absolute convergence if it is finite. Then if the power series (2) converges, the power series (1) also converges.

The convergence radius R refers to the boundary between the area the power series (2) converges absolutely and doesn’t. If \(|z|<R\), the power series (1) converges absolutely. Then if \(|z|>R\), it diverges. When \(|z|=R\), some power series converge, others diverge. Therefore, we need to think about converging or diverging for each power series.

How to Calculate the Convergence Radius

I’ll show you two methods of finding the convergence radius.

1.The way using an expression with the upper limit as shown below.

$$\limsup_{ n \to \infty } {\sqrt[n]{a_n}}=\frac{1}{R}・・・(3)$$

2.When the limit \(\displaystyle \lim_{n\to \infty} \frac{|a_{n+1}|}{|a_n|}\) exists, the following expression holds.

$$\displaystyle \lim_{ n \to \infty }\frac{|a_{n+1}|}{|a_n|}=\frac{1}{R}・・・(4)$$

In both methods, if the limit value is 0, the convergence radius becomes \(R=∞\). In other words, if the absolute value \(|z|\) is not infinity, the power series always converge.

For example, as shown in the example below, the convergence radius of \(e^z\) is \(R=∞\). Therefore, if \(|z|\) is not infinite, \(e^z\) always converges and is a finite value. In fact, if you draw a graph of \(e^z\), you can confirm that \(e^z\) is a finite value until \(z\) goes to infinity.

The first method requires deep knowledge of mathematics and it is troublesome, so this time we will solve the example in the second way.


Consider the convergence radius of the next \(f(z)\).

\begin{eqnarray} \displaystyle f(z)&=&e^z\\&=&1+z+\frac{z^2}{2!}+・・・\\&=&\sum_{n=0}^N \frac{z^n}{n!}\\&=&\sum_{n=0}^N a_nz^n\end{eqnarray}


Link:Taylor Expansion

We also say


Substitute this into the equation (4).

\begin{eqnarray} \displaystyle \lim_{ n \to \infty }\frac{|a_{n+1}|}{|a_n|}&=&\displaystyle \lim_{ n \to \infty }\frac{\left| \frac{1}{(n+1)!} \right|}{\left| \frac{1}{n!} \right|}\\&=&\displaystyle \lim_{ n \to \infty }\frac{n!}{(n+1)!}\\&=&\displaystyle \lim_{ n \to \infty }\frac{n・(n-1)・(n-2)・・・・}{(n+1)・n・(n-1)・(n-2)・・・・}\\&=&\displaystyle \lim_{ n \to \infty }\frac{1}{n+1}\\&=&0 \end{eqnarray}

Finally, the convergence radius is obtained.