# Binomial series

A power series of the form

$$\sum _ { n=0 } ^ \infty \left ( \begin{array}{c} \alpha \\ n \end{array} \right ) z ^ {n} = 1 + \left ( \begin{array}{c} \alpha \\ 1 \end{array} \right ) z + \left ( \begin{array}{c} \alpha \\ 2 \end{array} \right ) z ^ {2} + \dots ,$$

where $n$ is an integer and $\alpha$ is an arbitrary fixed number (in general, a complex number), $z = x + iy$ is a complex variable, and the

$$\left ( \begin{array}{c} \alpha \\ n \end{array} \right )$$

are the binomial coefficients. For an integer $\alpha = m \geq 0$ the binomial series reduces to a finite sum of $m + 1$ terms

$$(1+z) ^ {m} = 1 + mz + { \frac{m(m-1)}{2!} } z ^ {2} + \dots + z ^ {m} ,$$

which is known as the Newton binomial. For other values of $\alpha$ the binomial series converges absolutely for $| z | <1$ and diverges for $| z | > 1$. At points of the unit circle $| z | = 1$ the binomial series behaves as follows: 1) if $\mathop{\rm Re} \alpha > 0$, it converges absolutely at all points; 2) if $\mathop{\rm Re} \alpha \leq -1$, it diverges at all points; and 3) if $-1 < \mathop{\rm Re} \alpha \leq 0$, the binomial series diverges at the point $z = -1$ and converges conditionally at all other points. At all points of convergence, the binomial series represents the principal value of the function ${(1 + z) } ^ \alpha$ which is equal to one at $z = 0$. The binomial series is a special case of a hypergeometric series.

If $z = x$ and $\alpha$ are real numbers, and $\alpha$ is not a non-negative integer, the binomial series behaves as follows: 1) if $\alpha > 0$, it converges absolutely on $-1 \leq x \leq 1$; 2) if $\alpha \leq -1$, it converges absolutely in $-1 < x < 1$ and diverges at all other values of $x$; and 3) if $-1 < \alpha \leq 0$, the binomial series converges absolutely in $-1 < x < 1$, converges conditionally at $x = 1$, and diverges for $x = -1$; for $| x | > 1$ 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 " 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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binomial_series&oldid=46068
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article