Question

Prove: If \(g\) is a rational function defined for all nonnegative integers, then \(\sum a_{n} x^{n}\) and \(\sum a_{n} g(n) x^{n}\) have the same radius of convergence. HINT: Use Exercise \(4.1 .30(a)\).

Short answer

Question: Prove that if \(g\) is a rational function defined for all nonnegative integers, then the power series \(\sum a_{n} x^{n}\) and \(\sum a_{n} g(n) x^{n}\) have the same radius of convergence. Answer: This can be proved using the result from Exercise 4.1.30(a) by showing that \(\limsup_{n\to\infty}\frac{|a_{n}|^{\frac{1}{n}}}{|a_{n} g(n)|^{\frac{1}{n}}}=1\). We find that the limit superior equals 1 by considering the properties of rational functions and their polynomial components, and in turn, conclude that the original power series \(\sum a_{n} x^{n}\) and the transformed power series \(\sum a_{n} g(n) x^{n}\) have the same radius of convergence.

Step-by-step solution

Recall the definition of a rational function and Exercise 4.1.30(a)

A rational function is a function of the form \(g(x)=\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials in \(x\). We are looking to show that the power series \(\sum a_{n} x^{n}\) and \(\sum a_{n} g(n) x^{n}\) have the same radius of convergence. According to Exercise 4.1.30(a), we need to show: $$ \limsup_{n\to\infty}\frac{|a_{n}|^{\frac{1}{n}}}{|a_{n} g(n)|^{\frac{1}{n}}} = 1 $$ which will imply that the radius of convergence for both series are equal.

Show that the limit superior equals 1

We have: $$ \limsup_{n\to\infty}\frac{|a_{n}|^{\frac{1}{n}}}{|a_{n} g(n)|^{\frac{1}{n}}} = \limsup_{n\to\infty}\frac{1}{|g(n)|^{\frac{1}{n}}} $$ Since \(g(x)\) is a rational function, we can write it as: $$ g(x) = \frac{P(x)}{Q(x)} $$ where \(P(x)\) and \(Q(x)\) are polynomial functions. Hence, $$ \limsup_{n\to\infty}\frac{1}{|g(n)|^{\frac{1}{n}}} = \limsup_{n\to\infty}\frac{1}{\left|\frac{P(n)}{Q(n)}\right|^{\frac{1}{n}}} $$ Now we will consider the behaviors of \(P(x)\) and \(Q(x)\). We know that the growth of a polynomial is determined by the term with the highest degree. Let's suppose that \(P(x)\) has degree \(m\) and \(Q(x)\) has degree \(k\). Then, $$ \limsup_{n\to\infty}\frac{1}{\left|\frac{P(n)}{Q(n)}\right|^{\frac{1}{n}}} = \limsup_{n\to\infty}\frac{|Q(n)|^{\frac{1}{n}}}{|P(n)|^{\frac{1}{n}}} $$ Now, we know that the limit of the ratio of two polynomials is determined by the terms with the highest degrees. Therefore, $$ \limsup_{n\to\infty}\frac{|Q(n)|^{\frac{1}{n}}}{|P(n)|^{\frac{1}{n}}} = \limsup_{n\to\infty}\frac{n^{\frac{k}{n}}}{n^{\frac{m}{n}}} $$ Since \(g\) is a rational function, and \(g(n)\) is defined for all nonnegative integers, we must have that \(Q(n) \neq 0\) for any nonnegative integer \(n\). Therefore, the term with the highest degree of \(Q(x)\) must have the same exponent as the term with the highest degree of \(P(x)\). Thus, \(k=m\), meaning that their ratio is equal to 1.

Conclude the proof

We have shown that the limit superior of the ratio of the power series is equal to 1, meaning that the original power series \(\sum a_{n} x^{n}\) and the transformed power series \(\sum a_{n} g(n) x^{n}\) have the same radius of convergence.

Key concepts

Power Series

A power series is an infinite series of the form \( \sum a_n x^n \), where \( n \) is a non-negative integer and \( a_n \) represents the coefficient of each term. The power series is centered around a specific point, typically zero, and its convergence depends on the value of \( x \).

The radius of convergence is one of the essential features of a power series. It tells us the values of \( x \) for which the series converges, or sums up to a finite number.

• When the series converges for values of \( x \) within a certain boundary, we say that it converges with a positive radius.
• If it converges only at \( x = 0 \), the radius is zero.
• If it converges for all \( x \), the radius is infinite.

Understanding the radius of convergence is critical as it determines how far from the center point our calculations will remain accurate. Calculating this radius typically involves the use of the ratio or root tests.

Rational Function

A rational function is defined as the ratio of two polynomials. If we have two polynomial functions \( P(x) \) and \( Q(x) \), then a rational function \( g(x) \) can be written as \( g(x) = \frac{P(x)}{Q(x)} \).

Rational functions play a significant role in many areas of mathematics and are particularly interesting because they often simplify complex expressions and help make sense of polynomial behavior.

• They are continuous within their domain, except at points where the denominator \( Q(x) \) equals zero.
• The degree of \( P(x) \) and \( Q(x) \) often influences the behavior and the asymptotic nature of the rational function.

In our exercise context, rational functions are used to transform a power series whereby multiplying a sequence with a rational function preserves the same radius of convergence, provided it is defined for all non-negative integers.

Limit Superior

The limit superior, often denoted as \( \limsup \), is a fundamental concept in real analysis, primarily when dealing with sequences. It gauges the largest accumulation point of a sequence over its entire course. In simpler terms, it provides the highest level the terms of a sequence can approach as \( n \) gets large.

In the context of power series, the limit superior helps determine the radius of convergence of a power series by analyzing its terms.

• The term sequence \( \limsup_{n \to \infty} \frac{|a_n|^{1/n}}{|a_ng(n)|^{1/n}} \) helps analyze the convergence by equating it to 1 in the exercise.
• This comparison shows that the growth rates of the original series and the transformed series by a rational function are identical.

The use of limit superior, especially in assessing power series, provides insights that are pivotal for concluding convergence behaviors without needing to evaluate every term's limit separately.