Question

Suppose that \(k\) is a positive integer and $$ f(x)=\sum_{n=0}^{\infty} a_{n} x^{n} $$ has radius of convergence \(R\). Show that the series $$ g(x)=f\left(x^{k}\right)=\sum_{n=0}^{\infty} a_{n} x^{k n} $$ has radius of convergence \(R^{1 / k}\).

Short answer

To summarize, the radius of convergence of the series \(g(x) = f(x^k) = \sum_{n=0}^{\infty} a_{n} x^{k n}\), where \(f(x) = \sum_{n=0}^{\infty} a_{n} x^{n}\) has a radius of convergence \(R\), is given by \(R_g = R^{\frac{1}{k}}\).

Step-by-step solution

Define the given function and series

In the exercise, we are given that: $$ f(x)=\sum_{n=0}^{\infty} a_{n} x^{n} $$ with radius of convergence \(R\). We are also given another series: $$ g(x)=f\left(x^{k}\right)=\sum_{n=0}^{\infty} a_{n} x^{k n} $$ We need to find the radius of convergence of this series.

Find a formula for the radius of convergence

By definition, the radius of convergence of a power series is given by: $$ R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_{n}|}} $$ In this case, the radius of convergence of \(g(x)\) is denoted by \(R_g\). Let's try to find a formula for \(R_g\) using the given information.

Find the formula for \(R_g\)

In order to find \(R_g\), we need to find the limit in the formula above for the series \(g(x)\). For this series, we need to determine the limit of \(\sqrt[kn]{|a_n x^{kn}|}\). Using the properties of the limit and power series properties, we get: $$ \sqrt[kn]{|a_n x^{kn}|} = \sqrt[kn]{|a_n|} \sqrt[kn]{|x^{kn}|} = \sqrt[kn]{|a_n|} |x^{k}| $$ So, the limit as \(n \to \infty\) is: $$ \limsup_{n \to \infty} \sqrt[kn]{|a_n x^{kn}|} = |x^k| \limsup_{n \to \infty} \sqrt[kn]{|a_n|} $$ Now we can use this limit to find \(R_g\). By definition, we have: $$ R_g = \frac{1}{\limsup_{n \to \infty} \sqrt[kn]{|a_n x^{kn}|}} $$ Plugging in the limit we got above, we have: $$ R_g = \frac{1}{|x^k| \limsup_{n \to \infty} \sqrt[kn]{|a_n|}} $$

Relate \(R_g\) to given radius of convergence, \(R\)

We know the radius of convergence \(R\) for the original series is given by: $$ R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_{n}|}} $$ So, we can write the limit as: $$ \limsup_{n \to \infty} \sqrt[n]{|a_{n}|} = \frac{1}{R} $$ Let's plug this into our formula for \(R_g\): $$ R_g = \frac{1}{|x^k| \frac{1}{R}} $$ Now we can simplify to get: $$ R_g = \frac{R}{|x^k|} $$ Since we want the radius of convergence as an expression of \(R\), we can solve for \(x^k\) and get: $$ x^k = \frac{R}{R_g} $$ So, $$ |x^k| = \frac{R}{R_g} $$ Taking the \(k^{th}\) root of both sides, we get the final result: $$ R_g = R^{\frac{1}{k}} $$ This shows that the radius of convergence of the series \(g(x)\) is \(R^{1 / k}\).

Key concepts

Power Series

A power series is an infinite series of the form \( f(x) = \sum_{n=0}^{\infty} a_n x^n \), where \( a_n \) are coefficients and \( x \) is a variable. The power series represents a function as an infinite sum of terms involving powers of \( x \). Power series are crucial in mathematical analysis because they can approximate functions that are complex or otherwise difficult to work with.

Key properties of a power series include:

• **Convergence**: A power series may converge, that is, produce a finite value, for certain values of \( x \).
• **Radius of Convergence**: This is a value \( R \) such that the series converges if \( |x| < R \) and diverges if \( |x| > R \). For \( |x| = R \), convergence may depend on the series.
• **Term-by-term Differentiation and Integration**: Within the radius of convergence, a power series can be differentiated and integrated term by term.

Understanding power series allows us to work effectively with complex functions, breaking them into manageable parts via their coefficients. This dissection becomes invaluable when examining the behavior near points of interest, such as limits and singularities.

Limsup

Limsup, short for "limit superior," is an important concept in analysis, specifically when working with sequences. Given a sequence \( \{a_n\} \), the limsup captures the largest value that its subsequences approach at infinity. Mathematically, it is defined as the smallest limit point of the sequence, or formally:

\[ \limsup_{n \to \infty} a_n = \lim_{n \to \infty} \sup_{m \geq n} a_m \]

This quantity becomes particularly useful in determining the radius of convergence of a power series. By identifying\( \limsup_{n \to \infty} \sqrt[n]{|a_n|} \), for the series \( f(x) = \sum_{n=0}^{\infty} a_n x^n \), this method allows us to encapsulate all the upper tendencies of the series' terms. For the given series, the radius of convergence \( R \) is expressed as:

\[ R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}} \]

This formula highlights the power of the limsup as a tool to analyze and confirm the behavior of sequences that contribute to the evaluation of the convergence intervals of series.

Analytic Functions

Analytic functions play a crucial role in mathematical analysis and complex variables. These functions are defined by their power series representation and have derivatives of all orders in their domain. An analytic function can essentially be expressed or approximated as a power series, which is a defining characteristic of these functions.

Key traits of analytic functions include:

• **Infinite Differentiability**: They can be differentiated infinitely often, and these derivatives themselves give rise to the power series representation.
• **Local Representation**: Around every point within their radius of convergence, analytic functions can be expressed as a power series.
• **Holomorphic Nature**: Within the context of complex analysis, analytic functions are also called holomorphic when they are complex differentiable in a neighborhood.

The study of analytic functions often involves understanding their radius of convergence, as this defines the domain over which their power series approximation holds true. In exercise problems like the given one, recognizing how transformations of the variable (like substituting \( x^k \)) affect the radius of convergence requires an understanding of these fundamental properties and how they extend under power series operations.