The theorem that, for Integral Positive ,

the so-called Binomial Series, where are Binomial Coefficients. The theorem was known for the case by Euclid around 300 BC, and stated in its modern form by Pascal in 1665. Newton (1676) showed that a similar formula (with Infinite upper limit) holds for Negative Integral ,

the so-called Negative Binomial Series, which converges for .

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 10, 1972.

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 307-308, 1985.

Conway, J. H. and Guy, R. K. ``Choice Numbers Are Binomial Coefficients.'' In *The Book of Numbers.* New York:
Springer-Verlag, pp. 72-74, 1996.

Coolidge, J. L. ``The Story of the Binomial Theorem.'' *Amer. Math. Monthly* **56**, 147-157, 1949.

Courant, R. and Robbins, H. ``The Binomial Theorem.'' §1.6 in
*What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, pp. 16-18, 1996.

© 1996-9

1999-05-26