Problem 19 - Calculus: Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {(x - 2)^n}{n^n} \)

Step-by-Step Solution

Answer

The radius of convergence is infinite, and the series converges for all real \( x \).

Step 1: Identify the General Term

The general term of the series is given by \( a_n = \frac{(x-2)^n}{n^n} \). This expression will be used in the upcoming steps to determine the radius and interval of convergence.

Step 2: Apply the Ratio Test

The Ratio Test is used to find the radius of convergence. Calculate the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \): \[ a_{n+1} = \frac{(x-2)^{n+1}}{(n+1)^{n+1}} \, \text{and} \, a_n = \frac{(x-2)^n}{n^n} \]\[ \frac{a_{n+1}}{a_n} = \frac{(x-2)^{n+1}}{(n+1)^{n+1}} \cdot \frac{n^n}{(x-2)^n} = \frac{(x-2) \cdot n^n}{(n+1)^{n+1}} \]\[ = (x-2) \cdot \frac{n^n}{(n+1)^{n+1}} = (x-2) \cdot \left(\frac{n}{n+1}\right)^n \cdot \frac{1}{n+1} \]

Step 3: Simplify the Limit Expression

To further simplify, consider the limit:\[ L = \lim_{n \to \infty} (x-2) \cdot \left(\frac{n}{n+1}\right)^n \cdot \frac{1}{n+1} \]As \(n \to \infty\), \(\left(\frac{n}{n+1}\right)^n \to \frac{1}{e}\) and \(\frac{1}{n+1} \to 0\). Hence, the entire expression goes to zero, which satisfies the condition for the Ratio Test, \( L < 1 \). Thus, the series converges for all \(x\).

Step 4: State the Radius and Interval of Convergence

Since the Ratio Test gives \( L = 0 \) for all values of \( x \), the radius of convergence \( R = \infty \). Consequently, the interval of convergence is \((-\infty, \infty)\), meaning the series converges for all real numbers \( x \).

Key Concepts

Ratio Test

The Ratio Test is a straightforward method used to determine the convergence of an infinite series. It involves taking the limit of the absolute value of the ratio of consecutive terms of the series. For a series with terms given by \( a_n \), this test is applied as follows:

• Compute the limit: \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
• Interpretation: If \( L < 1 \), the series converges absolutely. If \( L > 1 \), the series diverges. If \( L = 1 \), the test is inconclusive.

In our exercise, we use the Ratio Test to calculate the series' limit, which helps in finding the radius of convergence for the power series. The limit ultimately approaches zero, indicating convergence for all values of \( x \). This conclusion implies a radius of convergence of infinity.

Interval of Convergence

The interval of convergence is a range of \( x \) values for which an infinite series converges. It's calculated from knowing the radius of convergence and examining the endpoints of the interval to see where the series converges. When using the Ratio Test, this interval is pivotal for determining where the series is valid.

• The series converges within the interval \((x_0 - R, x_0 + R)\), where \( R \) is the radius of convergence.
• It is important to check the series' behavior at the interval's endpoints individually, as convergence can sometimes fail there.

In our example, since the radius of convergence is infinite, the interval becomes \((-\infty, \infty)\). This indicates that the series converges for all real numbers, meaning no endpoints need extra checking.

Power Series

A power series is a series of the form \( \sum_{n=0}^{\infty} a_n (x - c)^n \), where each term involves a power of \( (x - c) \). Power series are incredibly useful in mathematical applications, including approximating functions, solving differential equations, and evaluating integrals.

• Each \( a_n \) represents the coefficient of the term \( (x-c)^n \).
• The series' center, \( c \), is the point around which the series is expanded.
• Convergence of a power series depends on the distance between \( x \) and \( c \), known as the radius of convergence.

Understanding these concepts is essential for determining where and how a series can be applied. In our exercise, the power series is centered at \( x = 2 \), with coefficients involving the factorial term \( n^n \), demonstrating the series' complexity yet wide convergence range.

Infinite Series

An infinite series is a sum of infinitely many terms, typically written as \( \sum_{n=0}^{\infty} a_n \). Converging infinite series yield a finite result, whereas diverging series do not settle on a single value.

• The convergence of such series can be tested using different mathematical strategies like the Ratio Test, Root Test, or Direct Comparison Test.
• Mathematicians often use infinite series for working with functions and solving problems that involve approximations.

Our exercise deals with a specific kind of infinite series—a power series. This example demonstrates how infinite series are fundamental in mathematical analysis, providing the basis for understanding complex behaviors and functions throughout different mathematical fields. In this case, the convergence holds for all real numbers, showcasing the robustness of power series.