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 $1+m x+m(m-1)x^2/(2\cdot 1) + \cdots$" 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)