Approximate the Sum of the Series

While the above approximate percentages add to only 099 the actual decimals are repeating so the sum would actual be a repeating series of 9s after the decimal. This implies that the function is analytic at every point of the interval or disk.

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A function may differ from the sum of its Taylor series even if its Taylor series is convergent.

. The power series method can be applied to certain nonlinear differential equations though with less flexibility. By definition this is the same as 1 or 100 percent. A very large class of nonlinear equations can be solved analytically by using the ParkerSochacki methodSince the ParkerSochacki method involves an expansion of the original system of ordinary differential equations through auxiliary equations.

A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval or open disk in the complex plane containing x.

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