L25 Representation of Functions as Power Series I

L25 Representation of Functions as Power Series I - a 3 ( x...

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Lecture 25: Representation of functions as Power Series (I) How do we evaluate such integral Z ln(1 ± t ) t dt ? How do we evaluate such integral Z 0 : 2 0 1 1 + x 5 dx ? Recall the Geometric Series 1 X n =0 x n = 1 1 ± x ; j x j < 1 Using this series, you can represent other functions as power series. 1
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Express the functions as power series: ex. 3 1 ± x 4 ex. x 4 x + 1 2
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ex. 3 x 2 + x ± 2 3
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Di±erentiation and Integration of Power Series You can di±erentiate and integrate power series term-by-term. The radius of convergence re- mains the same, but the interval of convergence could change. Theorem: If the power series P a n ( x ± a ) n has radius of convergence R > 0, then the func- tion f ( x ) = P 1 n =0 a n ( x ± a ) n = a 0 + a 1 ( x ± a )+ a 2 ( x ± a ) 2 + ²²² is di±erentiable on ( a ± R;a + R ), and 1). f 0 ( x ) = a 1 + 2 a 2 ( x ± a ) + 3
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Unformatted text preview: a 3 ( x a ) 2 + = 1 X n =1 na n ( x a ) n 1 2). Z f ( x ) dx = C + a ( x a ) + a 1 ( x a ) 2 2 + = C + 1 X n =0 a n ( x a ) n +1 n + 1 4 ex. (a). Find the power series representation of the function f ( x ) = ln(1 x ) : (b) Approximate ln 2 without a calculator. (excited yet?) 5 Power Series for Arctan x It can easily be shown that 1 1 + x 2 = 1 X n =0 ( 1) n x 2 n for 1 x 1 : 6 ex. (1)Find the power series representation for arctan x . (2) Evaluate arctan 1. Tonights homework: Write functions as power series. 7...
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L25 Representation of Functions as Power Series I - a 3 ( x...

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