Problem 6 - Complex Analysis

Question

Let \(\sum_{n=0}^{\infty} a_{n} z^{n}\) be a power series with radius of convergence \(r\) Show: (a) If \(R:=\lim _{n \rightarrow \infty}\left|a_{n}\right| /\left|a_{n+1}\right|\) exists, then \(r=R\). (b) If \(\tilde{\rho} \lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} \in[0, \infty]\) exists, then \(r=1 / \tilde{\rho} .\) Here we formally use the conventions \(1 / 0=\infty\) and \(1 / \infty=0 .(r=0\) for \(\tilde{\rho}=\infty\), and \(r=\infty\) for \(\tilde{\rho}=0 .)\) (c) If we set $$ \rho:=\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}:=\lim _{n \rightarrow \infty}\left(\sup \left\\{\sqrt[n]{\left|a_{n}\right|}, \sqrt[n+1]{\left|a_{n+1}\right|}, \sqrt[n+2]{\left|a_{n+1}\right|}, \ldots\right\\}\right), $$ then - following the same conventions as in (b)- one hasa $$ r=1 / \rho $$

Step-by-Step Solution

Answer

(a) \( r = R \); (b) \( r = \frac{1}{\tilde{\rho}} \); (c) \( r = \frac{1}{\rho} \).

Step 1: Understanding the Power Series

The power series given is \( \sum_{n=0}^{\infty} a_{n} z^{n} \). The series converges for \( |z| < r \), where \( r \) is the radius of convergence. The task is to find an expression for \( r \) using different limits involving the coefficients \( a_n \).

Step 2: Apply the Ratio Test for Part (a)

The ratio test for convergence of a series says that if \( L = \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| \) exists, then the series converges if \( L < 1 \) and diverges if \( L > 1 \). We are given \( R = \lim_{n \rightarrow \infty} \frac{|a_n|}{|a_{n+1}|} \), which implies \( L = \frac{1}{R} \). Hence, \( r = R \), as the series converges for \( |z| < R \).

Step 3: Use the Root Test for Part (b)

The root test states that for a series \( \sum a_n \), if \( \tilde{\rho} = \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|} \) exists, the series converges if \( \tilde{\rho} < 1 \) and diverges if \( \tilde{\rho} > 1 \). Therefore, \( r = \frac{1}{\tilde{\rho}} \). If \( \tilde{\rho} = \infty \), then \( r = \frac{1}{\infty} = 0 \), and if \( \tilde{\rho} = 0 \), then \( r = \frac{1}{0} = \infty \).

Step 4: Understand the Limit Definition for Part (c)

Here, \( \rho = \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|} \) is given as the limit of the supremum of \( n^{th} \) roots of coefficients. This is equivalent to the root test limit used in part (b). Thus, by the same logic, \( r = \frac{1}{\rho} \). If \( \rho = \infty \), then \( r = 0 \), and if \( \rho = 0 \), then \( r = \infty \).

Key Concepts

Radius of Convergence

The Radius of Convergence is a core concept in understanding the range within which a power series, such as \( \sum_{n=0}^{\infty} a_{n} z^{n} \), will converge. The radius \( r \) determines the values of \( z \) for which the series converges. It’s critically important when dealing with power series. Knowing \( r \) helps us identify whether our series will produce a finite result or not for a given \( z \).
In essence, if \( |z| < r \), the series converges; otherwise, it diverges. This range where the series converges is called the disk of convergence. The disk can be thought of as a 'safe zone' for evaluating power series. More technically, it can encompass all \( z \) values as long as they fall within this circle centered at the origin with radius \( r \).
Finding the radius can guide on whether calculations within the series will lead to meaningful and finite answers or not. It serves as a critical delimiter between convergence and divergence.

Ratio Test

The Ratio Test is a popular method in calculus for determining the convergence or divergence of infinite series, including power series. To apply it, we examine the limit \( L = \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| \). This test will tell us how terms in the series grow relative to one another as \( n \) becomes very large.

• If \( L < 1 \), then the series converges absolutely.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

In our problem, instead of directly using \( L \), we introduce \( R = \lim_{n \rightarrow \infty} \frac{|a_n|}{|a_{n+1}|} \), and obtain that \( L = \frac{1}{R} \). This relationship reveals that the series converges for \( |z| < R \).
The Ratio Test is particularly useful for series where each term is a rational function of the index \( n \), as it leverages the fact that growing ratios dictate convergence behavior.

Root Test

The Root Test is another powerful tool for assessing convergence of infinite series. It asks us to look at the \( n^{th} \) root of the absolute values of the series' terms: \( \tilde{\rho} = \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|} \). This test is especially effective for series where terms have exponential growth patterns.

• If \( \tilde{\rho} < 1 \), the series converges absolutely.
• If \( \tilde{\rho} > 1 \), the series diverges.
• When \( \tilde{\rho} = 1 \), the test does not provide definitive answers.

In the context of our exercise, if \( \tilde{\rho} \) is finite, the radius of convergence \( r \) is \( \frac{1}{\tilde{\rho}} \). If \( \tilde{\rho} = \infty \), then the series’ radius of convergence is 0; conversely, if \( \tilde{\rho} = 0 \), then \( r \) is infinite. These observations show how the growth pattern affects the capability of the series to converge.

Convergence Criteria

The convergence of a power series relies largely on the limits and tests we apply. We've looked at the Radius of Convergence, the Ratio Test, and the Root Test.
For a more rigorous understanding:

• Convergence criteria give us the tools necessary to evaluate the sum of infinite terms.
• The outcome of these tests - whether the terms grow slower than \( 1 \) - determines convergence.
• The ultimate goal is to establish an interval \( (\text{disk}) \) where the series approaches a limit.

We adopted conventions like \( \frac{1}{0} = \infty \) and \( \frac{1}{\infty} = 0 \) to handle special cases where convergence behavior becomes extreme. Such criteria encapsulate the behavior of a series as it expands indefinitely and help us predict whether our calculations within it will remain valid or eventually become meaningless.Understanding convergence criteria helps mathematicians and scientists to predict performance and ensure calculations are viable across complex domains.