Question

Find the disk of convergence for each of the following complex power series. $$\sum_{n=0}^{\infty} z^{n}$$

Short answer

The disk of convergence is \ \(|z| < 1\).

Step-by-step solution

Identify the General Term

The general term of the series is given by \ a_n z^n. Here, the series is \ \ \(a_n = 1 \ \), so the general term is \ \(z^n \).

Apply the Root Test for Convergence

For a series \ \(\sum_{n=0}^{\inty} a_n z^n \), the disk of convergence can be found using the Root Test. According to the Root Test, the radius of convergence \ (R) is given by: \ \ \[ \frac{1}{R} = \lim_{n \to \infty} \sqrt[n]{|a_n|}. \]

Determine the Radius of Convergence

Here, \ \(a_n = 1\ \).\ To find the radius of convergence, calculate: \ \[ \lim_{n \to\ \infty} \sqrt[n]{|1|} = \lim_{n \to \infty} 1^{1/n} = 1. \]

Find the Disk of Convergence

Since \ \( \frac{1}{R} = 1 \), the radius of convergence \ \(R\ \) is \ \(1 \). Therefore, the disk of convergence is: \ \ \[ |z| < 1 \].

Key concepts

Disk of Convergence

The Disk of Convergence is a critical area in the complex plane where a given power series converges. Understanding this region helps you know how far you can push the values of the complex variable within the series without causing divergence. In simple terms, when talking about a 'disk' here, imagine a circle centered at the origin (0,0) in the complex plane. The disk includes all points inside this circle.
For the power series \(\backslash sum_{n=0}^{\backslash infty} z^n\), the disk of convergence is the set of all points where the series converges. To find this disk, we first need to determine its radius of convergence. This is done using the Root Test.
The radius of convergence needs to be determined to accurately describe the 'disk'. Once you have the radius, you can simply state that the disk of convergence is \( \left|z\right| < R\), where \(R\) is the radius you calculated.

Radius of Convergence

The Radius of Convergence is the distance from the center of the disk of convergence to its boundary. This tells us how far from the center we can go before the power series stops converging. We use the Root Test for this purpose.
Consider this: for the series \(\backslash sum_{n=0}^{\backslash infty} z^n\), the general term \(a_n\) is simply 1 (because there’s no multiplying factor other than \(z^n\)). According to the Root Test formula: \[ \frac{1}{R} = \lim_{n \to \backslash infty} \sqrt[n]{\left|a_n\right|}. \] Applying this, we get: \[ \frac{1}{R} = \lim_{n \to \backslash infty} \sqrt[n]{\left|1\right|} = \lim_{n \to \backslash infty} 1^{1/n} = 1. \] Solving this, we find that \(R\ = 1.\) Thus, the radius of convergence for the series is 1. Therefore, the series will converge for all \( z \) within a distance of 1 from the center of the complex plane.

Root Test

The Root Test is a powerful tool to determine the radius of convergence of a complex power series. It helps us check if the series converges by investigating the n-th roots of the terms in the series.
Here’s the basic idea: you take the \(\backslash sqrt[n]{|a_n|}\) and look at what happens as \(n \rightarrow \backslash infty.\) Mathematically defined as: \[ \frac{1}{R} = \lim_{n \to \backslash infty} \sqrt[n]{\left|a_n\right|}. \] This limit gives us the inverse of the radius of convergence. If the limit exists and is finite, then its reciprocal is the radius of convergence. If the limit turns out to be zero, it means the radius of convergence is infinite, meaning the series converges everywhere in the complex plane. If, however, the limit is infinite, the series converges only at \( z = 0 \).
In our case, for the series \(\backslash sum_{n=0}^{\infty} z^n \), the term \(a_n \) is always 1. Thus: \[ \frac{1}{R} = \lim_{n \to \backslash infty} \sqrt[n]{1} = 1, \] Hence, \(R\ = 1. \)