A function defined by a power series with a radius of convergence R > 0 has a

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A function defined by a power seriesimage


with a radius of convergence R > 0 has a Taylor series that converges to the function at every point of (-R, R). Show this by showing that the Taylor series generated byimage


is the seriesimage itself.


An immediate consequence of this is that series likeimage


andimage


obtained by multiplying Taylor series by powers of x, as well as series obtained by integration and differentiation of convergent power series, are themselves the Taylor series generated by the functions they represent.

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Thomas Calculus Early Transcendentals

ISBN: 9780321884077

13th Edition

Authors: Joel R Hass, Christopher E Heil, Maurice D Weir

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