Problem 25 - Calculus: Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {5x - 4)^n}{n^3} \)

Step-by-Step Solution

Answer

Radius of convergence is 1; interval of convergence is \( \left[\frac{3}{5}, 1\right] \).

Step 1: Identify the Form of the Series

The series given is \( \sum_{n = 1}^{\infty} \frac{(5x - 4)^n}{n^3} \). This is a power series where the general term is \( a_n = \frac{(5x - 4)^n}{n^3} \). Its center is at \( x = \frac{4}{5} \).

Step 2: Apply the Root Test

For the root test, find the expression \( c = \lim_{n \to \infty} \sqrt[n]{\left| a_n \right|} \). In this case, \( \left| a_n \right| = \frac{\left| 5x - 4 \right|^n}{n^3} \). The root test gives, \( c = \lim_{n \to \infty} \sqrt[n]{\left| 5x - 4 \right|^n \frac{1}{n^3}} = \lim_{n \to \infty} \frac{\left| 5x - 4 \right|}{n^{3/n}} \approx \left| 5x - 4 \right| \).

Step 3: Determine the Radius of Convergence

The root test result \( c \) implies convergence when \( \left| 5x - 4 \right| < 1 \). Thus, the radius of convergence \( R \) is 1. The series converges when the distance to the center \( \frac{4}{5} \) from \( x \) is less than 1.

Step 4: Find the Interval of Convergence

Considering \( \left| 5x - 4 \right| < 1 \), solve \( -1 < 5x - 4 < 1 \). First solve for the left inequality: \( 5x - 4 > -1 \) gives \( 5x > 3 \), so \( x > \frac{3}{5} \). Then solve the right inequality: \( 5x - 4 < 1 \), giving \( 5x < 5 \), so \( x < 1 \). Combining these, the interval of convergence before checking endpoints is \( \left( \frac{3}{5}, 1 \right) \).

Step 5: Check the Endpoints

Test the endpoints \( x = \frac{3}{5} \) and \( x = 1 \). For \( x = \frac{3}{5} \), evaluate \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n^3} \), which converges by the Alternating Series Test. For \( x = 1 \), evaluate \( \sum_{n=1}^{\infty} \frac{1}{n^3} \), which converges by the p-series test (since \( p = 3 > 1 \)). Therefore, the interval of convergence, including the endpoints, is \( \left[ \frac{3}{5}, 1 \right] \).

Key Concepts

Radius of Convergence

The radius of convergence is one of the fundamental concepts when dealing with power series. It determines the range within which a power series converges and is crucial for understanding the behavior of the series.
For a power series of the form:
\[\sum_{n=0}^{\infty} a_n (x - c)^n\]the radius of convergence helps us to know how far from the center point \(c\), we can move along the x-axis while still having the series converge.

• The series converges absolutely when \(|x - c| < R\), where \(R\) is the radius of convergence.
• Beyond this radius, the series diverges.

Determining the radius can be done using tests like the Root Test, and it often involves finding the point at which the limit of terms of the series approaches a specified value or zero. Understanding and finding the radius of convergence is essential for determining where a power series reliably represents a function.

Interval of Convergence

Once you have identified the radius of convergence, the next step is to find the interval of convergence. This interval not only includes the core section where the absolute convergence criterion is met but also possibly the endpoints where different convergence tests might come into play.
The series converges within an interval determined by:

• If \(R\) is the radius and the center is \(c\), the interval is \((c - R, c + R)\).
• You must test the endpoints separately, as series may converge or diverge at these points based on different criteria such as the Alternating Series Test or the p-Series Test.

In the given exercise, after applying tests at the endpoints, the interval of convergence was found to be \(\left[\frac{3}{5}, 1\right]\). Including the endpoints depends on the specific convergence behavior of the series at those points.

Root Test

The Root Test is particularly useful for finding the radius of convergence. It's a convergence test that examines the nth root of the absolute value of the terms of the series.
The Root Test is applied as follows:
Calculate \( c = \lim_{n \to \infty} \sqrt[n]{|a_n|} \).

• If \(c < 1\), the series converges absolutely.
• If \(c > 1\), the series diverges.
• If \(c = 1\), the Root Test is inconclusive.

In our exercise, the Root Test was utilized to determine the radius of convergence where \(c \) was approximated around the single variable factor of the power series, helping to establish \(\left| 5x - 4 \right| < 1\), revealing a radius of 1. The Root Test is a powerful tool for handling series involving nth powers of terms.

Power Series

A power series is an infinite series of the form:
\[\sum_{n=0}^{\infty} a_n (x - c)^n\]where \(a_n\) represents the coefficient of the nth term and \(c\) is the center of the series. Power series are encountered frequently across many areas of mathematics due to their properties and usefulness in approximating functions.
Power series have many important properties:

• They can represent functions as infinite polynomials, making them useful in calculus and complex analysis for approximation.
• Convergence can only be assured in a particular interval, highlighted by the radius of convergence.
• Understanding interval specifics, as well as endpoint behavior, provides more depth in analyzing where the series effectively models a given function.

In the exercise, the given series is a power series centered at \(x = \frac{4}{5}\), and determining its radius and interval of convergence provides insights into its behavior, that is foundational for further analysis and application.