SpecialSeries - Special Series John E. Gilbert, Heather Van...

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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 proto-typical 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 term-by-term. 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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SpecialSeries - Special Series John E. Gilbert, Heather Van...

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