Binomial series

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A power series of the form

where is an integer and is an arbitrary fixed number (in general, a complex number), is a complex variable, and the

are the binomial coefficients. For an integer the binomial series reduces to a finite sum of terms

which is known as the Newton binomial. For other values of the binomial series converges absolutely for and diverges for . At points of the unit circle the binomial series behaves as follows: 1) if , it converges absolutely at all points; 2) if , it diverges at all points; and 3) if , the binomial series diverges at the point and converges conditionally at all other points. At all points of convergence, the binomial series represents the principal value of the function which is equal to one at . The binomial series is a special case of a hypergeometric series.

If and are real numbers, and is not a non-negative integer, the binomial series behaves as follows: 1) if , it converges absolutely on ; 2) if , it converges absolutely in and diverges at all other values of ; and 3) if , the binomial series converges absolutely in , converges conditionally at , and diverges for ; for the binomial series always diverges.

Binomial series were probably first mentioned by I. Newton in 1664–1665. An exhaustive study of binomial series was conducted by N.H. Abel [1], and was the starting point of the theory of complex power series.


[1] N.H. Abel, "Untersuchungen über die Reihe " J. Reine Angew. Math. , 1 (1826) pp. 311–339
[2] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)
[3] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Binomial series. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098