Section 10 6 Representations of functions as a power series Geometric series X n 0 xn 1 x 1 1 x Example 1 Find a power series representation for the function and determine the interval of convergence 1 1 1 x 2 1 1 x2 3 1 1 4x2 1 4 1 x2 1 x2 Term by term differentiation and integration If the power series P cn x a n has n 0 radius of convergence R 0 then the function f defined by 2 n f x c0 c1 x a c2 x a cn x a X cn x a n n 0 is differentiable and therefore continuous on the interval a R a R and 0 n 1 f x c1 2c2 x a ncn x a X ncn x a n 1 n 1 Z x a 2 x a n 1 cn f x dx C c0 x a c1 2 n 1 X x a n 1 cn C n 1 n 0 The radii of convergence of these series are R This does not mean that the interval of convergence remains the same Example 2 Find a power series representation for the function and determine the radius of convergence 1 1 1 x 2 2 2 ln 1 x Example 3 Evaluate an indefinite integral R tan 1 x2 dx as a power series Example 4 Use a power series to approximate the integral Z 1 2 tan 1 x2 dx 0 to six decimal places 3
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