AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
POWER SERIES
In the previous section we worked with the concept of approximating a function by using Taylor Polynomials. For example, the function f ()xe x can be approximated by its Maclaurin polynomials as follows:
xxxx2345 exx 1 2! 3! 4! 5!
REMEMBER: THE HIGHER THE DEGREE OF THE APPROXIMATION THE BEST THE APPROXIMATION WILL BE!!!!
Today we’ll study that several important functions, like ex , can be represented exactly by an infinite series called a power series.
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
Sample Problem #1:
Extend the following power series:
xn a) POWER SERIES CENTERED AT 0 n0 2!n
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
b) (1)(nnx 2) POWER SERIES CENTERED AT 2 n0
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
1 c) (3)x n POWER SERIES CENTERED AT -3 n0 n
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
RADIUS AND INTERVAL OF CONVERGENCE
A power series in x can be viewed as a function of x.
n f ()xaxc n n0 where the domain of f is the set of all x for which the power series converges. DETERMINATION OF THE DOMAIN OF A POWER SERIES IS THE PRIMARY CONCERN IN THIS SECTION.
n f (cc ) acan 0 so c ALWAYS LIE IN THE DOMAIN OF f . n0
STUDY TIP: To determine the radius of convergence of a power series, use the RATIO TEST.
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
Sample Problem #2:
Find the radius of convergence:
a) 2x n n0 AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
n 1 xn b) n n0 2
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
2!nx2n c) n0 n! AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
n x d) n0 5
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
ENDPOINT CONVERGENCE
NOTE that for a power series whose radius of convergence is a finite number R, the previous Theorem says nothing about the convergence at the endpoints of the interval of convergence. EACH ENDPOINT MUST BE TESTED SEPARETELY FOR CONVERGENCE OR DIVERGENCE. As a result there are six possible cases:
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
Sample Problem #3:
Find the interval of convergence for: Be sure to include a check for convergence at the endpoints of the interval. a) 11n1 nx n n0 AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
n nx!1 b) n1 1 3 5 ... 2n 1
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
DIFFERENTIATION AND INTEGRATION OF POWER SERIES
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
Sample Problem #4: Intervals of Convergence for f(x), its derivative, and its antiderivative
xn a) Consider the function fx() , find the intervals of convergence of f ()xdx, f ()x , f ' ()x n1 n
AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659
nn1 15x b) Consider the function fx() , find the intervals of convergence of n n1 n5 f ()xdx, f ()x , f ' ()x