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  2. Power series - Wikipedia

    en.wikipedia.org/wiki/Power_series

    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 ...

  3. Generating function - Wikipedia

    en.wikipedia.org/wiki/Generating_function

    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.

  4. Power series solution of differential equations - Wikipedia

    en.wikipedia.org/wiki/Power_series_solution_of...

    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.

  5. Padé approximant - Wikipedia

    en.wikipedia.org/wiki/Padé_approximant

    The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series , and it may still work where the Taylor series does ...

  6. List of mathematical series - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_series

    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.

  7. Binomial coefficient - Wikipedia

    en.wikipedia.org/wiki/Binomial_coefficient

    One may show by induction that F(n) counts the number of ways that a n × 1 strip of squares may be covered by 2 × 1 and 1 × 1 tiles. On the other hand, if such a tiling uses exactly k of the 2 × 1 tiles, then it uses n − 2 k of the 1 × 1 tiles, and so uses n − k tiles total.

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  9. Perturbation theory - Wikipedia

    en.wikipedia.org/wiki/Perturbation_theory

    In regular perturbation theory, the solution is expressed as a power series in a small parameter . [1] [2] The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the ...