11_08 Power Series.pdf - Math 1132Q Section 11.8 Power...

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Math 1132Q Section 11.8: Power Series Power Series : ( ) ( ) = - = 0 k k k c x b x f centered at c . Purpose: Approximate functions that can’t be represented as symbolically such as - - 1 1 sin , 2 dx x x dx e x , and the Bessel Function: ( ) ( ) ( ) = + + + - = 0 2 2 ! ! 2 1 k p k p k k p p k k x x J . Integrating and differentiating functions that can’t be represented symbolically Provides algorithm for computing values of functions such as ( ) x x e , sin , , π , etc. Examples: { } 1 , 1 1 0 < - = = x x x k k ( ) = = = - = 0 0 ! 0 ! 1 k k k k x k x x k e x y -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -2 0 2 4 6 8 10 Power functions must converge at x if they are to be used to approximate ( ) x f . Theorem: Given the power series: ( ) ( ) = - = 0 k k k c x b x f , there are exactly three possibilities: 1. The series converges for all x and the radius of convergence is = r Ratio test: 0 lim 1 = + n n n a a 2. The series converges only for c x = (diverging for c x ) and the radius of convergence is 0 = r Ratio test: = + n n n a a 1 lim 3. The series converges only for

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