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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, the center c is equal to zero, for instance for Maclaurin series.
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
which are particularly useful in the cases where the component sequence generating functions, (), can be expanded in a Laurent series, or fractional series, in , such as in the special case where all of the component generating functions are rational, which leads to an algebraic form of the corresponding diagonal generating function.
The exponential function e x for real values ... Here is a proof of Euler's formula using power-series ... Euler's formula provides a means of conversion between ...
There exist many types of convergence for a function series, such as uniform convergence, pointwise convergence, and convergence almost everywhere.Each type of convergence corresponds to a different metric for the space of functions that are added together in the series, and thus a different type of limit.
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).
In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent. The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics". [1]
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.