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11.8. Power Series. Power Series. A power series is a series of the form where x is a variable and the c n ’s are constants called the coefficients of the series. A power series may converge for some values of x and diverge for other values of x. Power Series.
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11.8 • Power Series
Power Series • A power series is a series of the form • where x is a variable and the cn’s are constants called the coefficients of the series. • A power series may converge for some values of x and diverge for other values of x.
Power Series • The sum of the series is a function • f(x) = c0 + c1x + c2x2 + . . . + cnxn + . . . • whose domain is the set of all x for which the series converges. Notice that f looks like a polynomial. The only difference is that f has infinitely many terms. • Also note: if we take cn = 1 for all n, the power series becomes the geometric series • xn = 1 + x + x2 + . . . + xn + . . . • which converges when –1 < x < 1 and diverges when | x | 1.
Power Series • More generally, a series of the form • is called a power series in (x – a) or a power series centered at a or a power series about a.
Radius of convergence and Interval of convergence • is • The Radius of Convergence for a power series is: The center of the series is x = a. The series converges on the open interval and may converge at the endpoints. • You must test each series that results at the endpoints of the interval separately for convergence. (We will practice this and it will make sense ) • Examples: The series is convergent on the interval [-3,-1] • but the series is convergent on the interval(-2,8].
Radius of convergence: R • The number R in case (iii) is called the radius of convergence of the power series. • This means: the radius of convergence is R = 0 in case (i) and R = in case (ii).
Interval of convergence • The interval of convergence of a power series is the interval that consists of all values of x for which the series converges. • In case (i) the interval consists of just a single point a. • In case (ii) the interval is ( , ). • In case (iii) note that the inequality |x – a| < R can be rewritten as a – R < x < a + R.
Radius and Interval of Convergence: • Examples:
Why study Power Series? • We will see that the main use of a power series is that it provides a way to represent some of the most important functions that arise in mathematics, physics, and chemistry. • Example: the sum of the power series, • , is called a Bessel function. • Many applications, mainly in: • EM waves in cylindrical waveguide, heat conduction • Electronic and signal processing • Modes of vibration of artificial membranes, Acoustics. • https://www.youtube.com/watch?v=ewIr4lO4408
The first few partial sums are • Graph of the Bessel function:
Representations of Functions as Power Series • We start with an equation that we have seen before: • We have obtained this equation by observing that the series is a geometric series with a = 1 and r = x. • But here our point of view is different. We now regard Equation 1 as expressing the function f(x) = 1/(1 – x) as a sum of a power series.
Example • Express 1/(1 + x2) as the sum of a power series and find the interval of convergence. • Solution: • Replacing x by –x2 in Equation 1, we have • Because this is a geometric series, it converges when |–x2| < 1, that is, x2 < 1, or |x| < 1.
Example – Solution • cont’d • Therefore the interval of convergence is (–1, 1). • (Of course, we could have determined the radius of convergence by applying the Ratio Test, but that much work is unnecessary here.)
Example: Approximation of sin(x) near x = a • (3rd order) • (1st order) • (5th order)
We use the previous facts to actually find the expansion of a function as a power series…. • Taylor and Maclaurin Series • 11.10
“Taylor” Series • Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series. • Brook Taylor • 1685 - 1731 • Greg Kelly, Hanford High School, Richland, Washington
If we make , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation. How does it work…? • Suppose we wanted to find a fourth degree polynomial of the form: • at • that approximates the behavior of
If we plot both functions, we see that near zero the functions match very well!
This pattern occurs no matter what the original function was! • Our polynomial: • has the form: • or:
Maclaurin Series: Taylor Series: (generated by f at ) (generated by f at ) Definition: If we want to center the series (and it’s graph) at zero, we get the Maclaurin Series:
Exercise 1: find the Taylor polynomial approximation at 0 (Maclaurin series) for:
Exercise 2: find the Taylor polynomial approximation at 0 (Maclaurin series) for: • Rather than start from scratch, we can use the function that we already know:
The 3rd order polynomial for is , but it is degree 2. • When referring to Taylor polynomials, we can talk about number of terms, order or degree. • This is a polynomial in 3 terms. • It is a 4th order Taylor polynomial, because it was found using the 4th derivative. • It is also a 4th degree polynomial, because x is raised to the 4th power. • The x3 term drops out when using the third derivative. • This is also the 2nd order polynomial.
. • Practice: Let’s work these now • 1) Show that the Taylor series expansion of ex is: • 2) Use the previous result to find the exact value of: • 3) Use the fourth degree Taylor polynomial of cos(2x) to find the exact value of
Convergence of Taylor Series: • is • If f has a power series expansion centered at x = a, then the • power series is given by • And the series converges if and only if the Remainder satisfies: • Where: is the remainder at x, (with c between x and a).
