Ad
related to: find a power series calculator from function
Search results
Results From The WOW.Com Content Network
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.
The exponential function (in blue), and the sum of the first n + 1 terms of its power series (in red) The exponential function is the sum of a power series: [2] [3] = + +! +! + = =!, where ! is the factorial of n (the product of the n first positive integers).
Formal power series are used in combinatorics to describe and study sequences that are otherwise difficult to handle, for example, using the method of generating functions. The Hilbert–Poincaré series is a formal power series used to study graded algebras.
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
An example is the function producing the x-th power of b from any real number x, where the base b is a fixed number. This function is written as f ( x ) = b x . When b is positive and unequal to 1, we show below that f is invertible when considered as a function from the reals to the positive reals.
Formal power series can be used to solve recurrences occurring in number theory and combinatorics. For an example involving finding a closed form expression for the Fibonacci numbers, see the article on Examples of generating functions. One can use formal power series to prove several relations familiar from analysis in a purely algebraic setting.