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Binomial Series

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There are several related series that are known as the binomial series.

The most general is

(x+a)^nu=sum_(k=0)^infty(nu; k)x^ka^(nu-k),
(1)

where (nu; k) is a binomial coefficient and nu is a real number. This series converges for nu>=0 an integer, or |x/a|<1 (Graham et al. 1994, p. 162). When nu is a positive integer n, the series terminates at n=nu and can be written in the form

(x+a)^n=sum_(k=0)^n(n; k)x^ka^(n-k).
(2)

The theorem that any one of these (or several other related forms) holds is known as the binomial theorem.

Special cases give the Taylor series

(1+x)^r=sum_(k=0)^(infty)((-r)_k)/(k!)(-x)^k
(3)
=1+rx+1/2r(r-1)x^2+1/6r(r-1)(r-2)x^3+....
(4)

where (r)_k is a Pochhammer symbol and |x|<1. Similarly,

(1-x)^(-r)=sum_(k=0)^(infty)((r)_k)/(k!)x^k
(5)
=1+rx+1/2r(r+1)x^2+1/6r(r+1)(r+2)x^3+...,
(6)

which is the so-called negative binomial series.

In particular, the case r=1/2 gives

(1-x)^(-1/2)=sum_(k=0)^(infty)((2k-1)!!)/((2k)!!)x^k
(7)
=sum_(k=0)^(infty)(-1)^k(-1/2; k)x^k
(8)
=1+1/2x+3/8x^2+5/(16)x^3+(35)/(128)x^4+...
(9)

(OEIS A001790 and A046161), where x!! is a double factorial and (n; k) is a binomial coefficient.

The binomial series has the continued fraction representation

(1+x)^n=1/(1-(nx)/(1+((1·(1+n))/(1·2)x)/(1+((1·(1-n))/(2·3)x)/(1+((2(2+n))/(3·4)x)/(1+((2(2-n))/(4·5)x)/(1+((3(3+n))/(5·6)x)/(1+...)))))))
(10)

(Wall 1948, p. 343).

See also

Binomial, Binomial Identity, Binomial Theorem, Multinomial Series, Negative Binomial Series

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 14-15, 1972.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Pappas, T. "Pascal's Triangle, the Fibonacci Sequence & Binomial Formula." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 40-41, 1989.Sloane, N. J. A. Sequences A001790/M2508 and A046161 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Binomial Series

Cite this as:

Weisstein, Eric W. "Binomial Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BinomialSeries.html

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