Binomial series
$$ 1. \ \frac {1}{1+x} = 1-x+x^2-x^3+\cdots, \ |x| \lt 1 $$
$$ 2. \ \frac {1}{1-x} = 1+x+x^2+x^3+\cdots, \ |x| \lt 1 $$
$$ 3. \ \sqrt {1+x}= 1+\frac x2 – \frac {x^2}{2\cdot 4} + \frac {1\cdot 3 x^3}{2\cdot 4 \cdot6} – \frac {1\cdot 3 \cdot 5 x^4}{2\cdot 4 \cdot6 \cdot 8} + \cdots , \ |x| \le 1 $$
$$ 4. \ \sqrt [3]{1+x}= 1+\frac x3 – \frac {1\cdot 2x^2}{3\cdot 6} + \frac {1\cdot 2\cdot 5 x^3}{3\cdot 6 \cdot9} – \frac {1\cdot 2\cdot 5 \cdot 8 x^4}{3\cdot 6 \cdot9 \cdot 12} + \cdots , \ |x| \le 1 $$
$$ 5. \ (1 − x)^{-2} = \frac {1}{(1- x)^2} = 1 + 2x + 3x^2 − 4x^3 +\cdots+ (r + 1)x^r + \cdots \infty $$
$$ 6. \ (1 + x)^{-2} = \frac {1}{(1+ x)^2} = 1 − 2x + 3x^2 − 4x^3 + \cdots + (−1)^r (r + 1)x^r + \cdots \infty $$
$$ 7. \ \frac {1}{\sqrt {1+x}} = 1-\frac 12 x+\frac 38 x^2-\frac {5}{16} x^3+ \frac {35}{128} x^4- \cdots $$
$$ 8. \ (1+x)^n= 1+nx+\frac {n(n-1)}{2!}x^2+\frac {n(n-1)(n-2)}{3!}x^3+\cdots $$
Example:
$$ \text{Expand} \ \sqrt {1+2x} $$
Solution:
$$ \sqrt {1+2x} = (1+2x)^{\frac 12} $$
$$ = 1+\left(\frac 12\right) (2x)+\frac {\left(\frac 12\right) \left(-\frac 12\right) }{2!}(2x)^2+\frac {\left(\frac 12\right) \left(-\frac 12\right) \left(-\frac 32\right)}{3!}(2x)^3+\cdots $$
$$ = 1+x-\frac 12 x^2 + \frac 12 x^3-\cdots $$