11.9 Representation of Functions as Power Series - Math...

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Math 1132 Worksheet 11.9 Name: Discussion Section: 11.9 Representations of Functions as Power Series Power Series, Derivatives, and Integrals. If the power series X n =0 c n ( x - a ) n has a radius of convergence R > 0, then the function f defined by f ( x ) = c 0 + c 1 ( x - a ) + c 2 ( x - a ) 2 + · · · = X n =0 c n ( x - a ) n is differentiable (and therefore continuous) on the interval ( a - R, a + R ) and (i) f 0 ( x ) = c 1 + 2 c 2 ( x - a ) + 3 c 3 ( x - a ) 2 + · · · = X n =1 nc n ( x - a ) n - 1 (ii) Z f ( x ) dx = C + c 0 ( x - a ) + c 1 ( x - a ) 2 2 + c 2 ( x - a ) 3 3 + · · · = C + X n =0 c n ( x - a ) n +1 n + 1 The radii of convergence of the power series in Equations (i) and (ii) are both R . Alternating Series Estimation Theorem. If s = ( - 1) n - 1 b n is the sum of an alternat- ing series that satisfies (i) b n +1 b n for all n (ii) lim n →∞ b n = 0 then | R n | = | s - s n | ≤ b n +1 .
Geometric Power Series. 1 1 - x = 1 + x + x 2 + x 3 + · · · = X n =0 x n | x | < 1 .

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