# Elementary Functions

## 1.1 Power of Binomials

### 1.11 Power series

1.110

${\left(1+x\right)}^{q}=1+qx+\frac{q\left(q-1\right)}{2!}{x}^{2}+\cdots +\frac{q\left(q-1\right)\dots \left(q-k+1\right)}{k!}{x}^{k}+\cdots ={\displaystyle \sum _{k=0}^{\infty}\left(\begin{array}{c}q\\ k\end{array}\right)}\text{?}{x}^{k}$

If q is neither a natural number nor zero, the series converges absolutely for |x| < 1 and diverges for |x| > 1. For x = 1, the series converges for q > −1 and diverges for q ≤ −1. For x = 1, the series converges absolutely for q > 0. For x = −1, it converges absolutely for q > 0 and diverges for q < 0. If q = n is a natural number, the series 1.110 is reduced to the finite sum 1.111. FI II 425

1.111

${\left(a+x\right)}^{n}={\displaystyle \sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right)}{x}^{k}{a}^{n-k}$

1.112

1.

${\left(1+x\right)}^{-1}=1-x+{x}^{2}-{x}^{3}+\cdots ={\displaystyle \sum _{k=1}^{\infty}{\left(-1\right)}^{k-1}{x}^{k-1}}$

(see also 1.121 2)

2.

${\left(1+x\right)}^{-2}=1-$

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