Problem 48 - Essential Calculus

Question

Write a power series that has the indicated interval of convergence. Explain your reasoning. (a) (-2,2) (b) (-1,1] (c) (-1,0) (d) [-2,6)

Step-by-Step Solution

Answer

The power series for each given interval are: (a) \( \sum_{n=0}^{\infty} x^{n} \), (b) \( \sum_{n=0}^{\infty} (-1)^{n}x^{n} \), (c) \( \sum_{n=0}^{\infty} (x+0.5)^{n} \), and (d) \( \sum_{n=0}^{\infty} (-1)^{n}(x-2)^{n} \)

Step 1: Solve for (a) (-2,2)

First, note that the interval (-2,2) has a length of 4, and hence the radius of convergence is 2. The series will be centered at 0. Considering that the convergence does not include the end points, we get the power series \( \sum_{n=0}^{\infty} x^{n} \), because it converges for |-x|<2 which implies -2

Step 2: Solve for (b) (-1,1]

For the interval (-1,1], the radius of convergence is 1 and the series is centered at 0. The convergence includes the endpoint 1, not -1. Thus, we can get the power series with \( \sum_{n=0}^{\infty} (-1)^{n}x^{n} \) because it converges for |-x|<=1 which implies -1

Step 3: Solve for (c) (-1,0)

For the interval (-1,0), the radius of convergence is 0.5 and the series is centered at -0.5. The convergence does not include the end points. Hence, the power series with \( \sum_{n=0}^{\infty} (x+0.5)^{n} \) is the solution, because |-x+0.5|<0.5 which implies -1

Step 4: Solve for (d) [-2,6)

The interval [-2,6) has a length of 8, hence the radius of convergence is 4 and centered at 2. The convergence includes the endpoint -2, not 6. We can use the power series \( \sum_{n=0}^{\infty} (-1)^{n}(x-2)^{n} \), as it covers the interval -2<=x<6.

Key Concepts

Interval of Convergence

The interval of convergence of a power series refers to the range of values of the variable for which the series converges, or sums up to a finite number. For a series centered at a certain point, the interval reveals how far from this center we can go while still having a valid power series. Let's break it down further:

• An interval can be open, closed, or a combination of both at its endpoints, which defines if the series converges at those endpoints.
• For instance, in the exercise's part (a) where the interval is (-2, 2), it's open, meaning that the power series will not converge when the variable equals -2 or 2.
• In part (b), we have an interval (-1, 1] which means the series converges up to and including the number 1 but excludes -1.

Understanding the interval of convergence is vital to know the applicable range of your series.

Radius of Convergence

The radius of convergence is a measure that tells us how far from the center point the power series converges. Imagine drawing a circle around the center of your interval; the radius of this circle is the radius of convergence.

• In the case of the exercise, part (a) has an interval from -2 to 2, giving it a radius of 2. This tells us that the series converges when the variable's distance from the center does not exceed this radius.
• For part (b), with a radius of 1, convergence happens up to that distance from the central point.
• The formula often used to find the radius, especially with more complex series, is the ratio test or root test, providing a practical approach to determining how far the series stretches.

The radius is always half the length of the interval.

Series Convergence

Series convergence refers to whether the sum of a series approaches a finite limit as more terms are added. For power series, this depends significantly on the value of the variable relative to the interval and radius of convergence.

• When the variable falls within the appropriate interval, the series converges, meaning the sum of its infinite terms levels off to a specific number.
• If a variable falls outside this interval, the series does not converge and can approach infinity or oscillate indefinitely.
• In the exercises, the specific formulas like \( \sum_{n=0}^{\infty} (-1)^{n}(x-2)^{n} \) or \( \sum_{n=0}^{\infty} x^{n} \) are chosen so that they converge precisely over their stated intervals.

Grasping convergence is crucial because it tells us when and where the series is valid.

Endpoints Inclusion

Endpoints inclusion stands for whether the endpoints of an interval are part of where the series converges. It's a detailed part of understanding the interval because including certain endpoints can drastically alter the convergence behavior.

• For example, part (b) where the interval is (-1, 1], includes the endpoint 1, meaning the power series converges not only within the range but exactly at x = 1.
• However, part (a)'s series does not include -2 or 2, so those exact values won't lead to a converging series.
• Also, note in part (d) with [-2, 6), the endpoint at -2 is included, but 6 is not. This precise distinction often requires checking endpoints separately using specific tests such as the Alternating Series Test or the Convergence Tests.

Whether an endpoint is included or not can be crucial to apply the series correctly.

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Key Concepts

Interval of Convergence

The interval of convergence of a power series refers to the range of values of the variable for which the series converges, or sums up to a finite number. For a series centered at a certain point, the interval reveals how far from this center we can go while still having a valid power series. Let's break it down further:

• An interval can be open, closed, or a combination of both at its endpoints, which defines if the series converges at those endpoints.
• For instance, in the exercise's part (a) where the interval is (-2, 2), it's open, meaning that the power series will not converge when the variable equals -2 or 2.
• In part (b), we have an interval (-1, 1] which means the series converges up to and including the number 1 but excludes -1.

Understanding the interval of convergence is vital to know the applicable range of your series.

Radius of Convergence

The radius of convergence is a measure that tells us how far from the center point the power series converges. Imagine drawing a circle around the center of your interval; the radius of this circle is the radius of convergence.

• In the case of the exercise, part (a) has an interval from -2 to 2, giving it a radius of 2. This tells us that the series converges when the variable's distance from the center does not exceed this radius.
• For part (b), with a radius of 1, convergence happens up to that distance from the central point.
• The formula often used to find the radius, especially with more complex series, is the ratio test or root test, providing a practical approach to determining how far the series stretches.

The radius is always half the length of the interval.

Series Convergence

Series convergence refers to whether the sum of a series approaches a finite limit as more terms are added. For power series, this depends significantly on the value of the variable relative to the interval and radius of convergence.

• When the variable falls within the appropriate interval, the series converges, meaning the sum of its infinite terms levels off to a specific number.
• If a variable falls outside this interval, the series does not converge and can approach infinity or oscillate indefinitely.
• In the exercises, the specific formulas like \( \sum_{n=0}^{\infty} (-1)^{n}(x-2)^{n} \) or \( \sum_{n=0}^{\infty} x^{n} \) are chosen so that they converge precisely over their stated intervals.

Grasping convergence is crucial because it tells us when and where the series is valid.

Endpoints Inclusion

Endpoints inclusion stands for whether the endpoints of an interval are part of where the series converges. It's a detailed part of understanding the interval because including certain endpoints can drastically alter the convergence behavior.

• For example, part (b) where the interval is (-1, 1], includes the endpoint 1, meaning the power series converges not only within the range but exactly at x = 1.
• However, part (a)'s series does not include -2 or 2, so those exact values won't lead to a converging series.
• Also, note in part (d) with [-2, 6), the endpoint at -2 is included, but 6 is not. This precise distinction often requires checking endpoints separately using specific tests such as the Alternating Series Test or the Convergence Tests.

Whether an endpoint is included or not can be crucial to apply the series correctly.