linguistic:translation, structural:reformatting
Binomial series

The binomial series is the power series

(1 + x) α = ∑ n = 0 ∞ (α n) x n {\displaystyle (1+x)^{\alpha }=\sum _{n=0}^{\infty }{\binom {\alpha }{n}}x^{n}}

whose coefficients are the generalized binomial coefficients

(α n) = ∏ k = 1 n α − k + 1 k = α (α − 1) ⋯ (α − n + 1) n !. {\displaystyle {\binom {\alpha }{n}}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}.}

(If n = 0, this product is an empty product and has value 1.) It converges for | x | < 1 {\displaystyle |x|<1} for any real or complex number α.

When α = −1, this is essentially the infinite geometric series mentioned in the previous section. The special cases α = 1 / 2 and α = − 1 / 2 give the square root function and its inverse:

(1 + x) 1 2 = 1 + 1 2 x − 1 8 x 2 + 1 16 x 3 − 5 128 x 4 + 7 256 x 5 − ⋯ = ∑ n = 0 ∞ (− 1) n − 1 (2 n) ! 4 n (n !) 2 (2 n − 1) x n, (1 + x) − 1 2 = 1 − 1 2 x + 3 8 x 2 − 5 16 x 3 + 35 128 x 4 − 63 256 x 5 + ⋯ = ∑ n = 0 ∞ (− 1) n (2 n) ! 4 n (n !) 2 x n. {\displaystyle {\begin{aligned}(1+x)^{\frac {1}{2}}&=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n-1}(2n)!}{4^{n}(n!)^{2}(2n-1)}}x^{n},\\(1+x)^{-{\frac {1}{2}}}&=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}}}x^{n}.\end{aligned}}}

When only the linear term is retained, this simplifies to the binomial approximation.