Not to be confused with IBM ThinkPad Power Series, The Power (TV series), Powers (American TV series), Powers (British TV series), or Power (TV series).
Infinite sum of monomials
In mathematics, a power series (in one variable) is an infinite series of the form
where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.
In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form
Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at 1⁄10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.
In mathematics, a powerseries (in one variable) is an infinite series of the form ∑ n = 0 ∞ a n ( x − c ) n = a 0 + a 1 ( x − c ) + a 2 ( x − c ) 2 +...
operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal powerseries is a special kind of formal series, whose terms...
mathematics, the Laurent series of a complex function f ( z ) {\displaystyle f(z)} is a representation of that function as a powerseries which includes terms...
the powerseries method is used to seek a powerseries solution to certain differential equations. In general, such a solution assumes a powerseries with...
Taylor series of a real or complex-valued function f (x), that is infinitely differentiable at a real or complex number a, is the powerseries f ( a )...
geometric series is a special case of the powerseries. The first term of a geometric series in expanded form is the coefficient a of that geometric series. In...
_{k=0}^{n}{\binom {n}{k}}x^{k}} holds for x = 0 only if 00 = 1. Similarly, rings of powerseries require x0 to be defined as 1 for all specializations of x. For example...
it is greater than one. It is particularly useful in connection with powerseries. The root test was developed first by Augustin-Louis Cauchy who published...
Power Racing Series is a non-profit power wheels racing competition. It's modeled after the racing series 24 Hours of LeMons however with child sized...
Lindelöf sum is a powerful method when applied to powerseries among other applications, summing powerseries in the Mittag-Leffler star. If g(z) is analytic...
Mighty Morphin Power Rangers (MMPR) is an American superhero television series that premiered on August 28, 1993, on the Fox Kids programming block. It...
In mathematics, characteristic powerseries may refer to: Multiplicative sequence Iwasawa algebra This disambiguation page lists mathematics articles associated...
radius of convergence of a powerseries is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative...
parallel to provide power from both motors simultaneously. Another operated in series with one source exclusively providing the power and the second providing...
The IBM ThinkPad PowerSeries (and subsequent IBM RS/6000 Model 860 Notebook) (800/820/821/822/823/850/851/860) is a laptop series from the ThinkPad line...
of numbers as the coefficients of a formal powerseries. Unlike an ordinary series, the formal powerseries is not required to converge: in fact, the generating...
also be differentiated using this rule. The power rule underlies the Taylor series as it relates a powerseries with a function's derivatives. Let f {\displaystyle...
both if some care is taken in preparation; newer chips in the Powerseries use the Power ISA. The history of RISC began with IBM's 801 research project...
Puiseux series are a generalization of powerseries that allow for negative and fractional exponents of the indeterminate. For example, the series x − 2...
Power Rangers Zeo is a television series and the fourth season of the Power Rangers franchise, based on the 19th Super Sentai series Chouriki Sentai Ohranger...
flat-faced cylinder and a cone are shapes that are members of the powerseries. The powerseries nose shape is generated by rotating the y = R(x/L)n curve about...
fundamental frequency. The kepstrum, which stands for "Kolmogorov-equation power-series time response", is similar to the cepstrum and has the same relation...
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