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Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...
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).
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.
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.
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.
Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line; Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola.
The Casio FX-7000G is a calculator which is widely known as being the world's first graphing calculator available to the public. It was introduced to the public and later manufactured between 1985 and c. 1988. [2] Notable features are its ability to graph functions, [3] and that it is programmable.