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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 .
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.
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.
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.
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).
The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section.
However, if the series is only known to be divergent, but for reasons other than diverging to infinity, then the claim of the theorem may fail: take, for example, the power series for +. At z = 1 {\\displaystyle z=1} the series is equal to 1 − 1 + 1 − 1 + ⋯ , {\\displaystyle 1-1+1-1+\\cdots ,} but 1 1 + 1 = 1 2 . {\\displaystyle {\\tfrac ...