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Alternatively, the equality can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of x 0 are equal to 0). Moreover, there can be no other power series with this property.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...
Since the Parker–Sochacki method involves an expansion of the original system of ordinary differential equations through auxiliary equations, it is not simply referred to as the power series method. The Parker–Sochacki method is done before the power series method to make the power series method possible on many nonlinear problems.
The first four partial sums of the series 1 + 2 ... is a simple consequence of the form of the series: ... 2 + 3 − 4 + ⋯ is the formal power series ...
The convergence criteria of the power series then apply, requiring ‖ ‖ to be sufficiently small under the appropriate matrix norm. For more general problems, which cannot be rewritten in such a way that the two matrices commute, the ordering of matrix products produced by repeated application of the Leibniz rule must be tracked.
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, ) approaches from below, even in cases where the radius of convergence, , of the power series is equal to and we cannot be sure whether the limit should be finite or not.
Thus, for the sake of clarity and readability, this Power Series wikipedia explanation needs a small edit to specify WHAT THING the power series "will converge" towards. I tried to make this simple improvement "A power series will converge towards an analytic function for some values of the variable x and may diverge for others" but it was ...