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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.
In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
In calculus, the reciprocal rule gives the derivative of the reciprocal of a function f in terms of the derivative of f.The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents.
In mathematics, a power series (in one variable) is an infinite series of the form = = + + + … where represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis , where they arise as Taylor series of infinitely differentiable functions .
With hindsight, however, it is considered the first general theorem of calculus to be discovered. [1] The power rule for differentiation was derived by Isaac Newton and Gottfried Wilhelm Leibniz, each independently, for rational power functions in the mid 17th century, who both then used it to derive the power rule for integrals as the inverse ...
In this case the algebra of formal power series is the total algebra of the monoid of natural numbers over the underlying term ring. [76] If the underlying term ring is a differential algebra, then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.
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