sec11_9

sec11_9 - (problem 22(1 Diﬀerentiate the function f x =...

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11.9 Representations of Functions as Power Series 1. Representing Functions using Power Series 1 1 - x =1+ x + x 2 + x 3 + ... = ± n =0 x n , | x | < 1 . Example 1.1. (problem 4) Find a power series for the function and determine the interval of convergence. f ( x )= 3 1 - x 4 Example 1.2. (problem 8) Find a power series for the function and determine the interval of convergence. f ( x )= x 2 x 2 +1 1

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Section 11.9 Representations of Functions as Power Series 2 2. Differentiating and Integrating Power Series Theorem 2.1. Let f ( x )= c n ( x - a ) n be a power series with radius of convergence R> 0 . Then (1) f is continuous on the interval ( a - R,a + R ) . (2) f is diﬀerentiable on the interval ( a - R,a + R ) and its derivative is f ± ( x )= d dx ± c n ( x - a ) n = ± d dx c n ( x - a ) n = ± nc n ( x - a ) n - 1 (3) f may be integrated on a closed interval contained in ( a - R,a + R ) and ² f ( x ) dx = ² ± c n ( x - a ) n dx = ± ² c n ( x - a ) n dx = ± c n n +1 ( x - a ) n +1 + C Example 2.1.
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Unformatted text preview: (problem 22) (1) Diﬀerentiate the function f ( x ) = tan-1 (2 x ) (2) Represent the derivative of the function f ( x ) as a power series. (3) Note the function f ( x ) is the integral of the function in part 2. Use integration to Fnd a power series representation of the function f ( x ) and the radius of convergence. Section 11.9 Representations of Functions as Power Series 3 Example 2.2. (problem 14) (1) Find a power series representation for f ( x ) = ln(1 + x ) . What is the radius of convergence? (2) Use part (1) to ±nd a power series for f ( x ) = x ln(1 + x ) . (3) Use part (1) to ±nd a power series for f ( x ) = ln( x 2 + 1) ....
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sec11_9 - (problem 22(1 Diﬀerentiate the function f x =...

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