Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The interval of convergence of a power series is \((-2,5]\). (a) What is the radius of convergence? (b) What is the center of the series?

Step-by-Step Solution

Answer

The radius of convergence is 3.5 and the center of the series is 1.5.

Step 1: Finding the Length of the Interval

First, find the length of the given interval. The interval is \((-2,5]\), so to find the length, subtract the left end of the interval from the right end of the interval, i.e., \(5 - (-2) = 7\).

Step 2: Finding the Radius of Convergence

The radius of convergence is half the length of the interval of convergence. Therefore, divide the obtained length by 2. So, \(7/2 = 3.5\)

Step 3: Finding the Center of the Series

The center of the series is just the midpoint of the interval of convergence, which can be found by averaging the endpoints of the interval, i.e., \((-2+5)/2 = 1.5\)

Key Concepts

Power Series

A power series is a type of infinite series that involves a variable raised to successive powers. It's expressed in the form \( \sum_{n=0}^{\infty} a_n (x - c)^n \), where \( a_n \) represents the coefficients, \( x \) is the variable of the series, and \( c \) is the center of the series. Power series are extremely versatile, as they can be used to represent various functions, especially in calculus for approximation purposes.

Some key characteristics of power series include:

• The terms include powers of \( x \), which means the series is dependent on a variable unlike typical numerical series.
• The series can converge or diverge depending on the value of \( x \).
• Power series can represent analytical functions within their interval of convergence.

Power series are pivotal in mathematical analysis as they provide tools for defining, manipulating, and approximating functions. Each specific power series has its own interval of convergence, centered around a particular point, which tells us the range of values for which the series will converge.

Radius of Convergence

The concept of the radius of convergence is central to understanding power series. It defines a distance within which the series will converge around the center. Our series might not converge for every value of \( x \), but within a certain radius from the center, it will.

To find the radius of convergence, especially if the interval is given, one needs to:

• Measure the total length of the interval of convergence.
• Divide this length by 2 to calculate the radius.

For example, if the interval of convergence is given as \((-2, 5]\), the length of the interval is \(7\). Divide this by 2, and the radius of convergence is \(3.5\). Thus, the series will converge at points \(3.5\) units away from the center. Understanding the radius of convergence helps in knowing where to safely use a power series to approximate a function accurately within a specified domain.

Center of the Series

The center of a power series is a key feature that helps in understanding its behavior and characteristics. It is the point \( c \) at which the series is centered and forms the basis from which the radius of convergence is measured outwards.

To find the center of the series, particularly when given the interval of convergence, you can calculate the midpoint of this interval. This is done by averaging the endpoints of the interval.
For instance, given the interval \((-2, 5]\), the center of the series can be determined by calculating the average: \((-2 + 5)/2 = 1.5\).
The center is crucial because it acts as the focal point around which convergence is measured. Knowing the center allows us to better understand where around this point the series might converge and therefore be used effectively in approximations and calculations.
In summary, the center influences both the convergence behavior and the usability of a power series to model functions within the determined radius.