Binomial series
A power series of the form
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where is an integer and
is an arbitrary fixed number (in general, a complex number),
is a complex variable, and the
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are the binomial coefficients. For an integer the binomial series reduces to a finite sum of
terms
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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.
References
[1] | N.H. Abel, "Untersuchungen über die Reihe ![]() |
[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) |
Binomial series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binomial_series&oldid=17445