POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to discuss is the power series and the use of this series. First of all, let us define what a power series. FACT: A power series about x = a is the series of the form The big question is when will a power series converge absolutely, converge conditionally, or diverge. To do this we will use a modified ratio test, and it will look like the following. Then you will have to determine where this expression is less than one. R is called the radius of convergence, and a - R < x < a + R is the interval of convergence. The power series converges absolutely for any x in that interval. Then we will have to test the endpoints of the interval to see if the power series might converge there too. If the series converges at an endpoint, we can say that it converges conditionally at that point. Any value outside this interval will cause the power series to diverge. EXAMPLE 1: First apply the modified ratio test remembering that x is a constant. 1 of 7
The radius of convergence for this series is 1. Now to determine the interval of convergence. x + 5 < 1-1 < x + 5 < 1-6 < x < -4 The interval of convergence is -6 < x < -4. This is where the series will converge absolutely. Now we must test the endpoints. This is a geometric series with r 1, therefore it diverges. This is a geometric series with r 1, therefore it diverges. Therefore, there are no values for which this series converges EXAMPLE 2: Again apply the modified ratio test and hold x constant. The radius of convergence for this series is 1. Now to determine the interval of convergence. 2 of 7
x + 2 < 1-1 < x + 2 < 1-3 < x < -1 Therefore, the interval convergence is -3 < x < -1. This is where this series will converge absolutely. Now we must test the endpoints. This is the harmonic series, and this series diverges. This is the alternating harmonic series, and this series converges. Therefore, this series converges conditionally at x = -1. EXAMPLE 3: Remember that x is a constant, so as n goes to infinity, the limit is zero. So this series converges absolutely for all x. The radius of convergence is and the interval of convergence is (-, ). EXAMPLE 4: 3 of 7
Since the limit is greater than 1, this series will only converge absolutely for x = 0. The radius of convergence for this series is 0. All other values of x will cause this power series to diverge. 1. There is a positive number R such that the series diverges for x - a > R, but converges absolutely for x - a < R. The series may or may not converge at either endpoints x = a - R and x = a + R. 2. The series converges absolutely for all x (R = ). 3. The series converges absolutely at x = a and diverges everywhere else (R = 0). EXAMPLE 5: Therefore, the radius of convergence is 4/3. Now to determine the interval of convergence. The interval of convergence is 4 of 7
and, this is where the series will converge absolutely. Now we must test the endpoints. This is a geometric series with r 1, therefore it diverges. This is a geometric series with r 1, therefore it diverges. There are no values for which this power series converges EXAMPLE 6: The radius of convergence is 2. x - 3 < 2-2 < x - 3 < 2 1 < x < 5 The interval of convergence is 1 < x < 5, and this is the interval where this series converges absolutely. Now to test the endpoints. 5 of 7
This is a geometric series with ratio r 1, therefore it diverges. This is a geometric series with r 1, therefore it diverges. There are no values for which this power series converges At the beginning of this set of supplemental notes, I stated that there are uses for the power series, now is the time to start discussing them. The main use of a power series is to numerically approximate integrals or derivative of functions that are not easily integrated or differentiated. We can do term-by-term differentiation of a power series of a function as long as x is inside the interval of convergence. This is also true for term-by-term integration. Consider the power series with x < 1. This is a geometric series with ratio less than one, so it sum is Now, let us integrate both sides of this equation. Let u = 1 - x, then du = -dx. Suppose, I wanted to find the derivative of 6 of 7
We will discuss this topic more after we discuss Taylor series. Power series will be very important in differential equations and other engineering courses, so I would make sure that I understand the topics that are associated with them. Work through these examples making sure to understand all of the steps. 7 of 7