China
Africa
Explore chapters and articles related to this topic
Sequences and series
Published in W. Bolton, Mathematics for Engineering, 2012
The term series is used for the sum of the terms of a sequence. There are many instances in mathematics were we have values which we can express as series and so aid calculations. A particular useful series is termed the binomial series (or binomial theorem) and is frequently used to simplify engineering expressions. In this chapter we consider the arithmetic, geometric and binomial series.
The binomial series
Published in John Bird, Higher Engineering Mathematics, 2017
The binomial series or binomial theorem is a formula for raising a binomial expression to any power without lengthy multiplication. The general binomial expansion of (a + x)n is given by: (a+x)n=an+nan−1x+n(n−1)2!an−2x200+n(n−1)(n−2)3!an−3x300+...
Behavioural modelling of delayed imbalance dynamics in nature: a parametric modelling for simulation of delayed instability dynamics
Published in International Journal of General Systems, 2022
Baris Baykant Alagoz, Furkan Nur Deniz, Murat Koseoglu
Binomial series has a long history; a special case of the Binomial theorem has been known by Euclid (Coolidge 1949). Binomial coefficients express the number of ways to select object groups when the order of objects in groups does not have a significance to form new groups. However, today, binomial expansion addresses the solution of various problems such as the approximate calculation of non-integer power terms in series expansion form. A formulation for this purpose can be written by where is a real number, is a positive integer, and is a Binomial coefficient which can be defined as
Power function and binomial series on
Published in Applied Mathematics in Science and Engineering, 2023
Seçil Gergün, Burcu Silindir, Ahmet Yantir
In the literature, one of the main advantages of power function is to establish the binomial series which has many physical applications. For this purpose, we intend to obtain the nabla -Taylor series of the nabla -power function (13), discuss its convergence and present nabla -binomial series. Since the nabla -power function is defined as a piecewise function in (13), we discuss its analyticity (namely -analyticity [15]) in two separate cases when and . When , the nabla -power function reduces to the ordinary power function Note that the classical power function converges to its binomial series, that is, provided that . Therefore, for , -power function converges to the binomial series The analyticity of the nabla -power function when will be presented in Theorem 4.6. First, let us recall the -Taylor series presented in Ref. [15]. For the rest of this section we assume .