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Binomial Series

For $\vert x\vert < 1$,

$\displaystyle (1+x)^n$ $\textstyle =$ $\displaystyle \sum_{k=0}^n {n\choose k} x^k$ (1)
  $\textstyle =$ $\displaystyle {n\choose 0}x^0+{n\choose 1}x^1+{n\choose 2}x^2+\ldots$ (2)
  $\textstyle =$ $\displaystyle 1+{n!\over 1!(n-1)!} x+{n!\over (n-2)!2!} x^2 +\ldots$ (3)
  $\textstyle =$ $\displaystyle 1+nx+{n(n-1)\over 2} x^2+\ldots.$ (4)

The binomial series also has the Continued Fraction representation
(1+x)^n={1\over\strut\displaystyle 1-{\strut\displaystyle nx...
...\strut\displaystyle x\over\strut\displaystyle 1+\ldots}}}}}}}.
\end{displaymath} (5)

See also Binomial Theorem, Multinomial Series, Negative Binomial Series


Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 14-15, 1972.

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.

© 1996-9 Eric W. Weisstein