Power Series

Power series for real numbers was covered in semester 1. This note explains the complex numbers version.

A complex power series centered at $\alpha$ is a series of the form

Here:

• $\alpha$: center, a fixed complex number
• $c_n$: coefficients, fixed complex numbers
• $z$: a complex variable

Radius of Convergence

The radius of convergence $\rho$ is defined as in the real case. For complex $z$, the interval of convergence becomes a disk.

There exists $\rho$ with $0 \le \rho \le \infty$ such that the series

• converges for $|z-\alpha| < \rho$
• diverges for $|z-\alpha| > \rho$

Ignoring boundary points, the convergence region is 1 of the following.

• Single point
  The point $\alpha$, when $\rho = 0$.
• Disk
  $D_\rho(\alpha) = \{z \in \mathbb{C} : |z-\alpha| < \rho\}$, when $0 < \rho < \infty$.
• Entire complex plane
  When $\rho = \infty$.

Examples:

• $\sum_{n=0}^{\infty} \frac{(n+1)(z-2)^n}{3^n}$ has $\rho = 3$ and disk of convergence $D_3(2)$

Algebra of Power Series

Let $f$ and $g$ have power series representations

Let $r = \min\{r_1, r_2\}$ and $\beta \in \mathbb{C}$. Then for $z \in D_r(\alpha)$:

• Scalar multiple
  $\beta f(z) = \sum_{n=0}^{\infty} \beta a_n (z-\alpha)^n$
• Sum
  $f(z) + g(z) = \sum_{n=0}^{\infty} (a_n + b_n)(z-\alpha)^n$
• Product
  $f(z)g(z) = \sum_{n=0}^{\infty} c_n (z-\alpha)^n$

The coefficients of the product are

This is the Cauchy product of the series.

Examples:

• The first 4 non-zero terms of the Maclaurin series of $e^z \sin z$ are $z + z^2 + \frac{z^3}{3} - \frac{z^5}{30}$

Worked Examples

Radius and Boundary

Find the disk of convergence of $\sum_{n=0}^{\infty} \frac{(n+1)(z-2)^n}{3^n}$.

Apply the ratio test.

The series converges for $\frac{|z-2|}{3} < 1$, so $\rho = 3$ and the disk is $D_3(2)$.

The ratio test is inconclusive on the boundary $|z-2| = 3$, since the limit equals $1$. Check the terms directly.

The $n$-th term is $u_n = \frac{(n+1)(z-2)^n}{3^n}$. Taking the modulus gives

On the boundary $|z-2| = 3$, so $\frac{|z-2|}{3} = 1$ and

A convergent series needs $u_n \to 0$. Here $|u_n|$ grows without bound, so the series diverges at every boundary point, regardless of which point on the circle $z$ is.

All 3 Regions

• $\sum n!\, z^n$
  The ratio is $(n+1)|z|$, which tends to $\infty$ for every $z \neq 0$. So $\rho = 0$ and the region is the point $0$.
• $\sum z^n$
  The ratio is $|z|$, so $\rho = 1$ and the region is $D_1(0)$.
• $\sum \frac{z^n}{n!}$
  The ratio is $\frac{|z|}{n+1} \to 0$ for every $z$. So $\rho = \infty$ and the region is the entire plane.

Sum of 2 Series

Let $f(z) = \frac{1}{1-z} = \sum z^n$ and $g(z) = \frac{1}{1+z} = \sum (-1)^n z^n$, both valid in $D_1(0)$.

The odd coefficients cancel and the even coefficients equal $2$. The series is valid in $D_1(0)$.

Cauchy Product of Exponential and Sine

Find the first 4 non-zero terms of the Maclaurin series of $e^z \sin z$.

The coefficients are

Apply $c_n = \sum_{k=0}^{n} a_k b_{n-k}$.

Cauchy Product with Geometric Series

Find the first 5 non-zero terms of the Maclaurin series of $\frac{e^z}{1-z}$.

Here $a_n = \frac{1}{n!}$ and $b_n = 1$, so $c_n = \sum_{k=0}^{n} \frac{1}{k!}$.

The series is valid in $D_1(0)$.