Problem 3 - Complex Analysis

Question

Give examples of power series with finite radius of convergence \(r \neq 0\), which have respectively one of the following properties: (a) the power series converges on the full boundary of the convergence disk, (b) the power series diverges on the full boundary of the convergence disk, (c) there are at least two convergence points and at least two divergence points on the boundary of the convergence disk.

Step-by-Step Solution

Answer

See specific examples: (a) Alternating harmonic series, (b) Squared index series, (c) Harmonic series.

Step 1: Understanding the Problem

We need to provide examples of power series with a finite radius of convergence, where the convergence behavior on the boundary of the convergence disk differs according to the given conditions.

Step 2: Example for Condition (a)

A power series that converges on the full boundary of the convergence disk is \[ \sum_{n=0}^{\infty} \frac{(-1)^n}{n+1}x^n \]. This series has a radius of convergence, say \( r = 1 \), and converges uniformly on the boundary \( |x| = 1 \). This is a well-known example that converges by the alternating series test.

Step 3: Example for Condition (b)

A power series that diverges on the full boundary of the convergence disk is \[ \sum_{n=0}^{\infty} x^{n^2} \]. This series converges inside the circle of radius \( r = 1 \), but on the boundary \( |x| = 1 \), the terms do not approach zero, causing divergence by the Cauchy-Hadamard theorem.

Step 4: Example for Condition (c)

A power series exhibiting both convergence and divergence on the boundary of the disk can be given as \[ \sum_{n=1}^{\infty} \frac{x^n}{n} \]. This series has a radius of convergence \( r = 1 \). It converges at \( x = 1 \) (by the harmonic series convergence), diverges at \( x = -1 \) (since \( \sum (-1)^n/n \) diverges), but at points like \( x = i \) and \( x = -i \), it can exhibit various behaviors.

Key Concepts

Radius of Convergence

The radius of convergence is a crucial concept when dealing with power series. It refers to the distance from the center of the series, within which the series converges absolutely. For a power series of the form \( \sum_{n=0}^{\infty} a_n (x - c)^n \), the radius of convergence, \( r \), is determined using the formula:
\[\frac{1}{r} = \limsup_{n \to \infty} \sqrt[n]{|a_n|}\]This means that if \(|x - c| < r\), the series converges absolutely. If \(|x - c| > r\), it diverges. For \(|x - c| = r\), the behavior can vary, which highlights the importance of understanding boundary behavior.

• Inside the disk: Series converges absolutely for all \(|x - c| < r\).
• On the boundary: Series behavior must be checked separately.

Convergence Disk

A convergence disk is a circular region in the complex plane centered around the center of the power series, with a radius equal to the radius of convergence. Within this disk, the power series converges absolutely. Think of it like a safety zone for convergence. If a power series has a radius of convergence \( r \), then the convergence disk is defined by the inequality \(|x - c| < r\).

• The interior of this disk contains all points where the series converges absolutely.
• Understanding the boundary points \(|x - c| = r\) is key, as convergence can vary.
• The convergence disk is one of the most important areas to analyze when studying series.

Boundary Behavior

Boundary behavior in the context of power series tends to be unpredictable at times. When a power series has a radius of convergence \( r \), the series converges for all \(|x - c| < r\); however, on the boundary where \(|x - c| = r\), the series may either converge or diverge, and the behavior can't be determined by the radius of convergence alone.
It often requires specific tests or examples to understand.

• Uniform convergence: Some series converge uniformly on their entire boundary, like \( \sum_{n=0}^{\infty} \frac{(-1)^n}{n+1} x^n \).
• Divergence examples: Some series diverge entirely on their boundary, such as \( \sum_{n=0}^{\infty} x^{n^2} \).
• Mixed behavior: Series can have mixed behavior, showing both convergence and divergence at different boundary points, as seen in \( \sum_{n=1}^{\infty} \frac{x^n}{n} \).

Alternating Series Test

The Alternating Series Test is a handy tool for establishing the convergence of a series with alternating terms, specifically those of the form \( \sum (-1)^n a_n \). This test is particularly useful due to its relative simplicity.
It states that a series \( \sum (-1)^n a_n \) will converge if all the following conditions are met:

• The terms \( a_n \) are positive.
• The terms \( a_n \) are decreasing: \( a_n > a_{n+1} \).
• The limit of \( a_n \) as \( n \to \infty \) is zero: \( \lim_{n \to \infty} a_n = 0 \).

This test simplifies the process of determining convergence, especially in cases like \( \sum_{n=0}^{\infty} \frac{(-1)^n}{n+1} x^n \), which fit the criteria perfectly. This underscores its application in practical convergence situations.