Determining The Radius Of Convergence Of Modified Power Series

In the fascinating realm of mathematical analysis, power series hold a position of paramount importance. These infinite series, expressed in the form

$\sum_{n=0}^{\infty} a_n (x - c)^n$,

where a_n represents the coefficients, x is the variable, and c denotes the center of the series, serve as a cornerstone for representing functions and solving differential equations. A crucial concept associated with power series is the radius of convergence, a non-negative real number or infinity that determines the interval within which the series converges. Understanding the radius of convergence is essential for determining the applicability and behavior of power series representations.

This article delves into an intriguing property of power series: how the radius of convergence transforms when the exponent of the variable x is modified. Specifically, we aim to prove that if the radius of convergence of the power series

$\sum_{n=1}^{\infty} u_n x^n$

is R, then the radius of convergence of the series

$\sum_{n=1}^{\infty} u_n x^{2n}$

is $\sqrt{R}$. This exploration not only deepens our understanding of power series but also provides valuable insights into manipulating and analyzing these fundamental mathematical objects.

Before embarking on the proof, let's briefly revisit the essential concepts that underpin our discussion:

• Power Series: An infinite series of the form $\sum_{n=0}^{\infty} a_n (x - c)^n$, where a_n are coefficients, x is a variable, and c is the center.
• Radius of Convergence (R): A non-negative real number or infinity that determines the interval of convergence for a power series. The series converges if |x - c| < R and diverges if |x - c| > R.
• Interval of Convergence: The set of all x values for which the power series converges.
• Ratio Test: A convergence test that states if lim_n→∞ |a_n+1 / a_n| = L, then the series converges if L < 1, diverges if L > 1, and the test is inconclusive if L = 1.
• Root Test: A convergence test that states if lim_n→∞ |a_n|^1/n = L, then the series converges if L < 1, diverges if L > 1, and the test is inconclusive if L = 1.

With these concepts firmly in place, we are well-equipped to tackle the proof at hand.

Theorem: If the radius of convergence of the power series $\sum_{n=1}^{\infty} u_n x^n$ is R, then the radius of convergence of the series $\sum_{n=1}^{\infty} u_n x^{2n}$ is $\sqrt{R}$.

Let's denote the first power series as $S_1(x) = \sum_{n=1}^{\infty} u_n x^n$ and the second power series as $S_2(x) = \sum_{n=1}^{\infty} u_n x^{2n}$. We are given that the radius of convergence of $S_1(x)$ is R. Our goal is to demonstrate that the radius of convergence of $S_2(x)$ is $\sqrt{R}$.

Step 1: Determining the Radius of Convergence of S1(x)

Since the radius of convergence of $S_1(x)$ is R, we can leverage the ratio test or the root test to express this mathematically. Let's employ the root test for this purpose. The root test states that if

$L = \lim_{n \to \infty} |u_n x^n|^{\frac{1}{n}} = \lim_{n \to \infty} |u_n|^{\frac{1}{n}} |x|$,

then the series converges if L < 1 and diverges if L > 1. Therefore, for $S_1(x)$ to converge, we must have

$\lim_{n \to \infty} |u_n|^{\frac{1}{n}} |x| < 1$,

which can be rewritten as

$|x| < \frac{1}{\lim_{n \to \infty} |u_n|^{\frac{1}{n}}}$.

Thus, the radius of convergence R for $S_1(x)$ is given by

$R = \frac{1}{\lim_{n \to \infty} |u_n|^{\frac{1}{n}}}$.

This implies that

$\lim_{n \to \infty} |u_n|^{\frac{1}{n}} = \frac{1}{R}$.

Step 2: Analyzing the Convergence of S2(x)

Now, let's turn our attention to the second power series, $S_2(x) = \sum_{n=1}^{\infty} u_n x^{2n}$. Again, we can apply the root test to determine its radius of convergence. For $S_2(x)$, the limit we need to consider is

$L' = \lim_{n \to \infty} |u_n x^{2n}|^{\frac{1}{n}} = \lim_{n \to \infty} |u_n|^{\frac{1}{n}} |x^2|$.

For $S_2(x)$ to converge, we require L' < 1, which translates to

$\lim_{n \to \infty} |u_n|^{\frac{1}{n}} |x^2| < 1$.

Step 3: Substituting and Simplifying

We know from Step 1 that $\lim_{n \to \infty} |u_n|^{\frac{1}{n}} = \frac{1}{R}$. Substituting this into the inequality above, we get

$\frac{1}{R} |x^2| < 1$,

which can be rearranged as

$|x^2| < R$.

Taking the square root of both sides (since both |x²| and R are non-negative), we obtain

$|x| < \sqrt{R}$.

Step 4: Concluding the Radius of Convergence

The inequality |x| < $\sqrt{R}$ precisely defines the region of convergence for $S_2(x)$. Therefore, the radius of convergence for the series $S_2(x) = \sum_{n=1}^{\infty} u_n x^{2n}$ is indeed $\sqrt{R}$.

In this article, we have successfully demonstrated that if the radius of convergence of the power series $\sum_{n=1}^{\infty} u_n x^n$ is R, then the radius of convergence of the modified power series $\sum_{n=1}^{\infty} u_n x^{2n}$ is $\sqrt{R}$. This result highlights an important relationship between the exponents of the variable x and the radius of convergence in power series. The proof relied on the fundamental concept of the radius of convergence, the root test, and algebraic manipulation. This exploration provides a deeper understanding of the behavior of power series and their convergence properties, which are crucial in various branches of mathematics, including complex analysis and differential equations. This theorem enriches our toolkit for analyzing and manipulating power series, contributing to a more comprehensive understanding of these essential mathematical constructs.

This exploration into the radius of convergence for modified power series opens up avenues for further investigation. One could explore how the radius of convergence changes when the exponent of x is altered to other forms, such as x³ⁿ, or even more generally, x^kn, where k is a positive integer. Additionally, investigating the behavior of power series when the coefficients u_n are modified or subjected to certain transformations could lead to further interesting results. These investigations could deepen our understanding of power series and their convergence properties, enhancing our ability to apply them in diverse mathematical contexts.