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Notes on Integrating and Differentiating Power Series

Notes on Integrating and Differentiating Power Series -...

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Notes on Differentiating and Integrating Power Series Since a power series can be considered as a function o its IOC, it is natural to ask: How does one perform certain operations on them that are customary on the function studied to date? These operations include differentiation and integration as well as certain arithmetic operations. Theorem : Suppose ( 29 ( 29 0 n n n f x c x a = = - has radius of convergence 0 R (or possibly R = ∞ ), Then f is differentiable (and therefore continuous) on ( 29 , R a R a - + and a) ( 29 ( 29 ( 29 1 0 1 n n n n n n d f x c x a nc x a dx - = = = - = - b) ( 29 ( 29 ( 29 1 0 0 1 n n n n n n c f x dx c x a dx x a K n + = = = - = - + + , where K is the constant of integration for the indefinite integral. c) In both cases the radius of convergence is R , the same as the ROC of the original series. Observations : There are several observations that one needs to keep in mind when applying the above theorem. Differentiating a series is accomplished by differentiating term by term. Integrating a series is accomplished by integrating term by term. While the radius of convergence is the same for all the series, the interval of convergence may be different. That is to say, the endpoints of the interval of convergence for the original series may or may not be included in the IOC for the derivative or the IOC for the integral. Examples : The following examples demonstrate how to apply the theorem in different circumstances. 1) Find a power series representation for the function ( 29 1 tan f x x - = and determine the interval of convergence. ( 29 1 2 0 tan 1 x dt f x x t - = = + (From Calculus I) ( 29 2 0 1 x dt t = - - (Recall the Geometric series 0 1 1 n n s s = = - and set 2 s t = - ) ( 29 ( 29 ( 29 2 2 2 0 0 0 0 0 0 1 1 x x x n n n n n n n n t dt t dt t dt = = = = - = - = - ( 29 ( 29 2 1 2 1 0 0 0 1 1 2 1 2 1 x n n n n n n t x n n + + = = - = - = + + .
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To determine the IOC for this power series, we know that the radius of convergence
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