Problem 31 - Calculus: Early Transcendentals

Question

If \( k \) is a positive integer, find the radius of convergence of the series \( \sum_{n = 0}^{\infty} \frac {(n!)^k}{(kn)!} x^n \)

Step-by-Step Solution

Answer

The radius of convergence is \( \infty \).

Step 1: Identify the Ratio Test Formula

To find the radius of convergence, we will use the Ratio Test. The formula for the Ratio Test is: \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|, \]where \( a_n \) is the general term of the series. If the limit is \( L < 1 \), then the series converges. If \( L > 1 \) or \( L = \infty \), the series diverges. If \( L = 1 \), the test is inconclusive.

Step 2: Write the General Term of the Series

The given series is \[ \sum_{n=0}^{\infty} \frac{(n!)^k}{(kn)!} x^n. \]The general term \( a_n \) is \( \frac{(n!)^k}{(kn)!} x^n \).

Step 3: Find the Ratio \( \frac{a_{n+1}}{a_n} \)

Calculate the term \( a_{n+1} \) as: \[ a_{n+1} = \frac{((n+1)!)^k}{((k(n+1))!)} x^{n+1}. \]Then find the ratio:\[ \frac{a_{n+1}}{a_n} = \frac{((n+1)!)^k}{((k(n+1))!)} x^{n+1} \cdot \frac{(kn)!}{(n!)^k x^n}. \]Simplify this expression:

Step 4: Simplify the Ratio and Take the Limit

Simplify \( \frac{a_{n+1}}{a_n} \) to:\[ \frac{(n+1)^k}{k(n+1) \cdot k(n+1) - 1 \cdot \ldots \cdot kn + 1} x. \]Taking the limit as \( n \to \infty \), note the factorial grows faster than polynomial or exponential terms, yielding:\[ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{(n+1)^k x}{(kn+k)!} = 0 \ ext{when} \ n \to \infty. \]

Step 5: Determine the Radius of Convergence

The limit \( L = 0 \) from the previous step implies the series converges for any \( x \). Therefore, the radius of convergence \( R \) is:\[ R = \infty. \]

Key Concepts

Ratio Test

The Ratio Test is a powerful tool in determining the convergence or divergence of a series. It involves analyzing the limit of the ratio of successive terms in the series. More specifically, we evaluate:\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]where \(a_n\) represents the general term of the series.
Here's how it works:

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

By using the Ratio Test, mathematicians can efficiently check the behavior of complex series, especially those involving powers or factorial terms. In the case of determining radii of convergence for power series, this test is invaluable.

Factorial Growth

Factorials describe the product of all positive integers up to a certain number. They grow very quickly as the number increases, much faster than simple powers or even exponential functions. This rapid growth is crucial in understanding series behavior.

• For instance, \( n! = 1 \times 2 \times 3 \times \ldots \times n \), growing substantially with larger \( n \).
• The rapid growth of factorial terms means when comparing factorials, larger \( n \) values dwarf polynomial or exponential terms.

In calculations, such as the Ratio Test in our series example, understanding factorial growth helps predict which components of the fraction will dominate as \( n \) increases, often simplifying the limit to 0 or a finite number.

Series Convergence

Determining if a series converges is about understanding whether adding up its infinite terms gives a finite number. Factors like the terms' size and behavior at large \( n \) influence this.

• If the terms get smaller fast enough as \( n \) increases, the series might converge.
• Conversely, if terms don't dwindle, the series could diverge.
• Tests like the Ratio Test assess these conditions efficiently by examining term ratios.

In our problem, the terms of the series involving factorial growth decrease swiftly due to the nature of factorials, suggesting a quick path to convergence for all \( x \). The outcome of our calculation, a radius of convergence \( R = \infty \), confirms the series converges for any real number \( x \).

Power Series

A power series is an infinite series that features terms of the form \(a_n x^n\) where \(a_n\) are coefficients and \(x\) is a variable. Power series are important because they can represent functions dynamically within a certain radius of convergence.

• They take the general form \( \sum_{n=0}^{\infty} a_n x^n \).
• The radius of convergence \( R \) is a key value indicating the interval in which the series converges.
• Understanding the convergence of such series helps in approximating functions and solving equations analytically.

In the exercise, the series analyzed has coefficients defined by factorials, influencing the convergence characteristics. The derived infinite radius of convergence suggests that this particular power series represents its function for all real \( x \). Such findings hint at the series' robust ability to model behavior across an extensive domain.