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Unformatted text preview: Special Series John E. Gilbert, Heather Van Ligten, and Benni Goetz As before lets start with a power series a + a 1 x + a 2 x 2 + a 3 x 3 + . . . + a n x n + . . . = summationdisplay n = 0 a n x n centered at x = 0 . On its Interval of Convergence it defines a function f ( x ) . For certain special series we can identify that function. The prototypical example is always the Geometric series f ( x ) = 1 + x + x 2 + x 3 + . . . + x n + . . . = summationdisplay n = 0 x n = 1 1 x . Its Interval of Convergence is ( 1 , 1) . in Lecture 7 we saw that algebraic tricks often enabled us obtain other representations from it such as y 2 3 + y = summationdisplay n =0 ( 1) n y n +2 3 n +1 = y 2 3 y 4 3 2 + y 4 3 3 + . . . + y n +2 3 n +1 + . . . . Its Interval of Convergence is ( 3 , 3) . But we are learning calculus, so it seems natural to try to use differentiation or integration as suggested already at the beginning of Lecture 7. A fundamental question arises immediately: weve used many times the fact that the derivative of a FINITE sum is the sum of the derivatives and that the integral of a FINITE sum is the sum of the integrals . In other words, we differentiate or integrate termbyterm. The crucial question is whether the same is true for infinite sums. YES, it is! This is where the Radius of Integration becomes important again: Important Properties: a power series n a n x n can be differentiated and...
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 Spring '11
 Gilbert
 Power Series

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