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Negative Binomial Series

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The series which arises in the binomial theorem for negative integer -n,

(x+a)^(-n)=sum_(k=0)^(infty)(-n; k)x^ka^(-n-k)
(1)
=sum_(k=0)^(infty)(-1)^k(n+k-1; k)x^ka^(-n-k)
(2)

for |x|<a.

For a=1, the negative binomial series simplifies to

(x+1)^(-n)=1-nx+1/2n(n+1)x^2-1/6n(n+1)(n+2)x^3+....
(3)

See also

Binomial Series, Binomial Theorem

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Cite this as:

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

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