
# 8.2: Convergence of Power Series


When we include powers of the variable $$z$$ in the series we will call it a power series. In this section we’ll state the main theorem we need about the convergence of power series. Technical details will be pushed to the appendix for the interested reader.

##### Theorem $$\PageIndex{1}$$

Consider the power series

$f(z) = \sum_{n = 0}^{\infty} a_n (z - z_0)^n.$

There is a number $$R \ge 0$$ such that:

1. If $$R > 0$$ then the series converges absolutely to an analytic function for $$|z - z_0| < R$$.
2. The series diverges for $$|z - z_0| > R$$. $$R$$ is called the radius of convergence. The disk $$|z - z_0| < R$$ is called the disk of convergence.
3. The derivative is given by term-by-term differentiation
$f'(z) = \sum_{n = 0}^{\infty} na_n (z - z_0)^{n - 1}$
The series for $$f'$$ also has radius of convergence $$R$$.
4. If $$\gamma$$ is a bounded curve inside the disk of convergence then the integral is given by term-by-term integration
$\int_{\gamma} f(z)\ dz = \sum_{n = 0}^{\infty} \int_{\gamma} a_n (z - z_0)^n$
##### Note
• The theorem doesn’t say what happens when $$|z - z_0| = R$$.
• If $$R = \infty$$ the function $$f(z)$$ is entire.
• If $$R = 0$$ the series only converges at the point $$z = z_0$$. In this case, the series does not represent an analytic function on any disk around $$z_0$$.
• Often (not always) we can find $$R$$ using the ratio test.
Proof

The proof of this theorem is in the appendix.

## Ratio Test and Root Test

Here are two standard tests from calculus on the convergence of infinite series.

##### Ratio Test

Consider the series $$\sum_{0}^{\infty} c_n$$. If $$L = \lim_{n \to \infty} |c_{n + 1}/c_n|$$ exists, then:

1. If $$L < 1$$ then the series converges absolutely.
2. If $$L > 1$$ then the series diverges.
3. If $$L = 1$$ then the test gives no information.
##### Note

In words, $$L$$ is the limit of the absolute ratios of consecutive terms.

Again the proof will be in the appendix. (It boils down to comparison with a geometric series.)

##### Example $$\PageIndex{1}$$

Consider the geometric series $$1 + z + z^2 + z^3 + ...$$. The limit of the absolute ratios of consecutive terms is

$L = \lim_{n \to \infty} \dfrac{|z^{n + 1}|}{|z^n|} = |z|$

Thus, the ratio test agrees that the geometric series converges when $$|z| < 1$$. We know this converges to $$1/(1 - z)$$. Note, the disk of convergence ends exactly at the singularity $$z = 1$$.

##### Example $$\PageIndex{2}$$

Consider the series $$f(z) = \sum_{n = 0}^{\infty} \dfrac{z^n}{n!}$$. The limit from the ratio test is

$L = \lim_{n \to \infty} \dfrac{|z^{n + 1}|/(n + 1)!}{|z^n|/n!} = \lim \dfrac{|z|}{n + 1} = 0.$

Since $$L < 1$$ this series converges for every $$z$$. Thus, by Theorem 8.3.1, the radius of convergence for this series is $$\infty$$. That is, $$f(z)$$ is entire. Of course we know that $$f(z) = e^z$$.

##### Test

Consider the series $$\sum_{0}^{\infty} c_n$$. If $$L = \lim_{n \to \infty} |c_n|^{1/n}$$ exists, then:

1. If $$L < 1$$ then the series converges absolutely.
2. If $$L > 1$$ then the series diverges.
3. If $$L = 1$$ then the test gives no information.
##### Note

In words, $$L$$ is the limit of the $$n$$th roots of the (absolute value) of the terms.

The geometric series is so fundamental that we should check the root test on it.

##### Example $$\PageIndex{3}$$

Consider the geometric series $$1 + z + z^2 + z^3 + ...$$. The limit of the $$n$$th roots of the terms is

$L = \lim_{n \to \infty} |z^n|^{1/n} = \lim |z| = |z|$

Happily, the root test agrees that the geometric series converges when $$|z| < 1$$.

8.2: Convergence of Power Series is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.