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

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

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
   \[f'(z) = \sum_{n = 0}^{\infty} na_n (z - z_0)^{n - 1} \nonumber \]
   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
   \[\int_{\gamma} f(z)\ dz = \sum_{n = 0}^{\infty} \int_{\gamma} a_n (z - z_0)^n \nonumber \]

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

\[L = \lim_{n \to \infty} \dfrac{|z^{n + 1}|}{|z^n|} = |z| \nonumber \]

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

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

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

\[L = \lim_{n \to \infty} |z^n|^{1/n} = \lim |z| = |z| \nonumber \]

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