Understand the definition of a power series. Find the radius and interval of convergence of a power series. Determine the endpoint convergence of a power series. Differentiate and integrate a power series. Objectives
We’ll strip you of any fears, drill you on the essentials, plug you in, make everything plane, and hopefully electrify you!
We’ve seen power series before… Does the name “Maclaurin” ring a bell?
Power Series An important function f(x) = e x can be represented exactly by an infinite series called a power series. For example, the Maclaurin series we found is a power series representation for e x For each real number x, it can be shown that the infinite series converges to the number e x.
Example 1 – Power Series a.The following power series is centered at 0. b.The following power series is centered at –1. c. The following power series is centered at 1.
Power Series A power series can be viewed as a function of x. The domain of a power series is the set of all x for which the power series converges. Every power series converges at its center because So c always lies in the domain of f.
Radius and Interval of Convergence
Power Series The domain of a power series can be a single point, an interval centered at c, or the entire real line.
The only tools we have to find said interval of convergence are the Ratio and Root Tests.
Example 2 – Finding the Radius of Convergence Find the radius of convergence of Solution: (use the Ratio Test) Therefore, by the Ratio Test, the series diverges for |x| > 0 and converges only at its center, 0. So, the radius of convergence is R = 0.
Why is the interval of convergence symmetric?
We have already studied a subset of all Power Series.
A power series whose radius of convergence is a finite number R, says nothing about the convergence at the endpoints of the interval of convergence. Each endpoint must be tested separately for convergence or divergence. As a result, the interval of convergence of a power series can take any one of the six forms shown in Figure Figure 9.18
Example 3 – Finding the Interval of Convergence Find the interval of convergence of Solution (using the Ratio Test):
Example 3 – Solution So, by the Ratio Test, the radius of convergence is R = 1. Moreover, because the series is centered at 0, it converges in the interval (–1, 1). This interval, however, is not necessarily the interval of convergence. To determine this, you must test for convergence at each endpoint. When x = 1, you obtain the divergent harmonic series cont'd
When x = –1, you obtain the convergent alternating harmonic series So, the interval of convergence for the series is [–1, 1): cont'd Example 3 – Solution
Ex: #4 Find the interval of convergence for the series. Since we are using a test that is only valid for positive series, we must put absolute value around any quantity that might be negative. Since the Root Test fails if the limit equals 1, we test the endpoints separately.
Ex: #5 If you get tired, it’s no wonder. We just did infinitely many problems twice!
Ex: #6 Now check the 2 endpoints. ; very divergent! ; also divergent by nth Term Test
Ex: #7 Check the endpoints. The radius of convergence is 1/2.
Ex: #28 Check the endpoints.
Ex: Determine the radius and interval of convergence: Series converges Series diverges
Ex: Determine the radius and interval of convergence: Radius of Convergence = 4 Test fails (where L=1) Need to test for convergence at endpoints of interval. Power series diverges at each endpoint. Interval of convergence is
Ex: Determine the radius and interval of convergence:
Just 1 more example!
Ex: Find the interval of convergence Rats! We have to use the Ratio Test when there is a factorial!
Ex: #20 (read note in book) Evidently, checking the endpoints is beyond the scope of this course as the text said: “don’t bother checking…the series diverges at the endpoints.”
Homework Day 1: Page 666: 1-35 odd Day 2: MMM
Differentiation and Integration of Power Series
Example 8 – Intervals of Convergence for f(x), f'(x), and ∫f(x)dx Consider the function given by Find the interval of convergence for each of the following. a. ∫f(x)dx b. f(x) c. f'(x)
Example 8 – Solution By Theorem 9.21, you have and By the Ratio Test, you can show that each series has a radius of convergence of R = 1. Considering the interval (–1, 1) you have the following. cont'd
Example 8(a) – Solution For ∫f(x)dx, the series converges for x = ±1, and its interval of convergence is [–1, 1 ]. See Figure 9.21(a). Figure 9.21(a) cont'd
Example 8(b) – Solution For f(x), the series converges for x = –1, and diverges for x = 1. So, its interval of convergence is [–1, 1). See Figure 9.21(b). Figure 9.21(b) cont'd
Example 8(c) – Solution For f'(x), the series diverges for x = ±1, and its interval of convergence is (–1, 1). See Figure 9.21(c). Figure 9.21(c) cont'd
EXAMPLE Determine the radius and interval of convergence for the power series