Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the radius of convergence of $$ \sum_{n=1}^{\infty} \frac{1 \cdot 2 \cdot 3 \cdots n}{1 \cdot 3 \cdot 5 \cdots(2 n-1)} x^{2 n+1} $$

Step-by-Step Solution

Answer

The radius of convergence is infinite.

Step 1: Identify the general term

First, identify the general term of the series: \[a_n = \frac{1 \cdot 2 \cdot 3 \cdots n}{1 \cdot 3 \cdot 5 \cdots(2 n-1)} x^{2 n+1} = \frac{n!}{(2n-1)!!} x^{2n+1}\]Here, \((2n-1)!!\) stands for the double factorial of the sequence of odd numbers up to \((2n-1)\).

Step 2: Apply the ratio test

To find the radius of convergence, apply the ratio test, which involves calculating the limit \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).Write down the term \(a_{n+1}:\)\[a_{n+1} = \frac{(n+1)!}{(2n+1)!!} x^{2n+3}\]Now compute the ratio:\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)! \cdot x^{2n+3}}{(2n+1)!!} \cdot \frac{(2n-1)!!}{n! \cdot x^{2n+1}} = \frac{(n+1)}{(2n+1)(2n)} x^2 \]This simplifies to:\[ \frac{n+1}{4n^2 + 2n} x^2 \]

Step 3: Compute the limit for the ratio test

Now compute the limit as \(n\) approaches infinity:\[\lim_{n \to \infty} \frac{n+1}{4n^2 + 2n} x^2 = \lim_{n \to \infty} \frac{1}{4n} x^2 = 0 \]The limit simplifies to 0 for any fixed \(x\).

Step 4: Conclude using the ratio test

Since the ratio test result is 0 for any \(x\), the radius of convergence is infinite.This means the series converges for all real values of \(x\) without restriction.

Key Concepts

Ratio Test

The Ratio Test is a popular mathematical tool that helps us determine if an infinite series converges or diverges. To apply the Ratio Test, we examine the limit of the ratio of consecutive terms in the series. Specifically, for a series with general term \(a_n\), we calculate:\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]If this limit is less than 1, the series converges absolutely. If it's greater than 1, or if it diverges, then the series does not converge. If the limit equals 1, the test is inconclusive, and other methods might be needed to determine convergence. In our exercise, when the result of the Ratio Test approached 0, it indicated that the radius of convergence of the power series was infinite, meaning the series converges for any value of \(x\).

Power Series

A power series is an infinite series of the form:\[ \sum_{n=0}^{\infty} c_n (x-a)^n \]where \(c_n\) represents the coefficients, and \(a\) is the center of the series. Power series are vital in expressing functions as sums of polynomial terms, particularly when the function does not have a simple expression. The convergence of a power series, specifically the values of \(x\) for which the series converges, is determined by its radius of convergence. The Radius of Convergence tells us the distance from the center \(a\) within which the series converges. In the exercise, solving for the radius of convergence involved using a ratio test which revealed that our series converges everywhere on the real line since the radius is infinite.

Double Factorial

The concept of a double factorial is an extension of the regular factorial operation. While the factorial of a non-negative integer \(n\), denoted \(n!\), is the product of all positive integers up to \(n\), the double factorial \((2n-1)!!\) for an integer \(n\) applies only to odd numbers leading up to \((2n-1)\). For example:- When applied to an odd number like 5, \(5!! = 5 \cdot 3 \cdot 1\) - For even numbers, like 4, it may apply the definition \((4)!! = 4 \cdot 2\). Double factorials frequently appear in combinatorial calculations and in series such as Legendre polynomials. In this exercise, the terms of the power series involve double factorials, affecting the structure of each term in the series but not complicating the application of the ratio test.

Infinite Series

An infinite series is the summation of an infinite sequence of terms. Infinite series, such as the one in our exercise, are crucial in mathematics for representing and working with functions that cannot be easily described by finite sums. There are several types of infinite series, and their convergence is a major topic in mathematical analysis. Key points about infinite series:

• If the sum of the terms approaches a specific value, the series converges.
• If not, the series diverges.
• Tests, such as the Ratio Test, can determine convergence.

In our exercise, we focus on finding the radius of convergence, which refers to the extent the series \(\sum a_n x^n\) remains convergent when \(x\) is bounded by a given radius from a center point. The outcome of our task reveals that for any \(x\), the power series converges, showing its broad applicability and convergence range.