AP BC Section 9.8: POWER , pg. 659

POWER SERIES

In the previous section we worked with the concept of approximating a by using Taylor . 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 n0 2!n

AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659

 b) (1)(nnx 2) POWER SERIES CENTERED AT 2 n0

AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659

 1 c)  (3)x  n POWER SERIES CENTERED AT -3 n0 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  n0 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 . n0

STUDY TIP: To determine the of a power series, use the .

AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659

Sample Problem #2:

Find the radius of convergence:

 a) 2x n n0 AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659

n  1 xn b)  n n0 2

AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659

 2!nx2n c)  n0 n! AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659

n  x d)  n0 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 . 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) 11n1 nx n n0 AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659

n  nx!1 b)  n1 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 , and its

 xn a) Consider the function fx()  , find the intervals of convergence of  f ()xdx, f ()x , f ' ()x n1 n

AP CALCULUS BC Section 9.8: POWER SERIES, pg. 659

nn1  15x  b) Consider the function fx() , find the intervals of convergence of  n n1 n5  f ()xdx, f ()x , f ' ()x