Power Series

10.7 Power Series 69 Power Series are sum of infinite polynomials. Def A power series about x=a has the form n 'in (x-a) = c o + '(x-a) + + + n=o where Co, C1 C21 Cn, constant coefficients and a is the center. A power series about X= O is then given by 8 cnx" = cn + + . . n 2 { (1) n=o Exp If Co=C = Cn=-= in (1) / then we get the geometric power & series X n = + x + 2 + X n = I if -1

Exp Find the radius of convergence and the interval of convergence for the following power series: 70 8 n-1 1 (-1) X Apply Ratio Test to Elvil n n=1 nt' n n I Un+1 lim X . = /xl lim = lim = nt X -DOO n-> 8 Un n as Thus, the radius of convergence is R=1. To find the interval of convergence: X The series converges absolutely for -1

E (x-2)" Apply Ratio Test 71 lim I Unti 4 Unti I = lim 10" Vn / (x-2) 10 . as 10 (x-2)" n=o = 1x-2/

00, : Thee If anx" converges absolutely on /X/< R , then 72 n=o aN D an [FCA]] converges absolutely on /FW/KR Gfor any continuous function f. 00, n Exp X = converges absolutely for 1x1

Th (Term by Term Integration Theorem) 73 00 suppose that f(x)= { cn (x-a)" converges for 1x-a/KR. n=o 00 00 n+) Then, E Cn (x-a) converges for /x-a)