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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 ...
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A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
A power series is a series of the form = (). The Taylor series at a point of a function is a power series that, in many cases, converges to the function in a neighborhood of . For example, the series
The power series method will give solutions only to initial value problems (opposed to boundary value problems), this is not an issue when dealing with linear equations since the solution may turn up multiple linearly independent solutions which may be combined (by superposition) to solve boundary value problems as well. A further restriction ...
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A formal series is a R-valued function c, on the free monoid A *, which may be written as ∑ w ∈ A ∗ c ( w ) w . {\displaystyle \sum _{w\in A^{*}}c(w)w.} The set of formal series is denoted R A {\displaystyle R\langle \langle A\rangle \rangle } and becomes a semiring under the operations