Question

Find the disk of convergence for each of the following complex power series. $$\sum_{n=1}^{\infty} \frac{z^{n}}{\sqrt{n}}$$

Short answer

Radius of convergence is 1. Disk: \(\{|z| < 1\}\).

Step-by-step solution

- Recognize the form of the series

The given series is \(\frac{z^{n}}{\frac{}{n}}\), which is a complex power series of the form \(\sum_{n=1}^{\infty}a_{n}z^{n}\).

- Identify the general term

For this series, the general term \(a_n\) is \(\frac{1}{\sqrt{n}}\).

- Apply the Root Test

To find the radius of convergence, apply the Root Test. First, find \(\lim_{n\to\infty}|a_n|^{1/n}\). In this case, \(a_n = \frac{1}{\sqrt{n}}\).

- Calculate the limit

Evaluate \(|a_n| = \frac{1}{\sqrt{n}}\). Then, calculate \(\lim_{n\to\infty}\left(\frac{1}{\sqrt{n}}\right)^{1/n}\).

- Simplify the expression

Simplify the expression inside the limit: \(\left(\frac{1}{\sqrt{n}}\right)^{1/n} = \left(n^{-1/2}\right)^{1/n} = n^{-1/(2n)}\).

- Determine the limit as n approaches infinity

As \(n \to \infty,\) \(n^{-1/(2n)} \to 1\) because \(\lim_{n\to\infty}\frac{1}{2n} = 0\).

- Compute the radius of convergence

According to the Root Test, the radius of convergence \(R\) is \(\frac{1}{\lim_{n\to\infty}|a_n|^{1/n}}\). Since \(\lim_{n\to\infty}|a_n|^{1/n} = 1\), \(R = 1\).

- State the disk of convergence

The series converges within the disk \(\{|z| < 1\}\).

Key concepts

disk of convergence

In a complex power series, the disk of convergence is the set of all points in the complex plane where the series converges.
The boundary of this disk is defined by the radius of convergence.
For a series centered at a point \(a\) in the complex plane, the disk of convergence is given by \(|z - a| < R\), where \(R\) is the radius of convergence.

radius of convergence

The radius of convergence of a complex power series is the distance from the center of the series to the boundary of the disk where the series converges.
This boundary is where the series starts to diverge. In other words, it’s the largest value of \(|z - a|\) for which the series converges.
To find this radius, we often use tests like the Root Test or Ratio Test. For the given series, we calculated the radius \(R\) and found it to be 1 using the Root Test.

root test

The Root Test is one method to determine the radius of convergence for a power series.
Given a series \(\textstyle\textsum_{n=0}^{\textinfty}a_n z^n\), apply the Root Test by finding the following limit: \(\textlim_{n\to\textinfty}|a_n|^{1/n}\).
If this limit is \(L\), the series converges if \(|z - a| < 1/L\). Conversely, it diverges if \(|z - a| > 1/L\).
In this exercise, the Root Test helped us find that \(R = 1\).

complex analysis

Complex analysis is a branch of mathematics focusing on functions of complex variables.
It is beneficial for understanding power series, including concepts like convergence, radius of convergence, and disks.
The tools and techniques used in complex analysis provide powerful methods to solve problems involving complex functions and series.