Huan Ma's Homepage

Power Series
$(1 \pm x)^{m} = 1 \pm m x + \frac{m (m - 1)}{2!} x^{2} \pm \frac{m (m - 1) (m - 2)}{3!} x^{3} + \dots + (\pm 1)^{n} \frac{m (m - 1) \dots (m - n + 1)}{n!} x^{n} + \dots$	$(\| x \| \leq 1)$
$(1 \pm x)^{- m} = 1 \mp m x + \frac{m (m + 1)}{2!} x^{2} \mp \frac{m (m + 1) (m + 2)}{3!} x^{3} + \dots + (\mp 1)^{n} \frac{m (m + 1) \dots (m + n - 1)}{n!} x^{n} + \dots$	$(\| x \| < 1)$
$(1 \pm x)^{\frac{1}{2}} = 1 \pm \frac{1}{2} x - \frac{1 \cdot 1}{2 \cdot 4} x^{2} \pm \frac{1 \cdot 1 \cdot 3}{2 \cdot 4 \cdot 6} x^{3} - \frac{1 \cdot 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 8} x^{4} \pm \dots$	$(\| x \| \leq 1)$
$(1 \pm x)^{- \frac{1}{2}} = 1 \mp \frac{1}{2} x + \frac{1 \cdot 3}{2 \cdot 4} x^{2} \mp \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} x^{3} + \frac{1 \cdot 3 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 6 \cdot 8} x^{4} \mp \dots$	$(\| x \| < 1)$
$(1 \pm x)^{- 1} = 1 \mp x + x^{2} \mp x^{3} + x^{4} \mp \dots$	$(\| x \| < 1)$
$(1 \pm x)^{- 2} = 1 \mp 2 x + 3 x^{2} \mp 4 x^{3} + 5 x^{4} \mp \dots$	$(\| x \| < 1)$
$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots + (- 1)^{n} \frac{x^{2 n + 1}}{(2 n + 1)!} + \dots$	$(\| x \| < \infty)$
$\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots + (- 1)^{n} \frac{x^{2 n}}{(2 n)!} + \dots$	$(\| x \| < \infty)$
$e^{x} = 1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots + \frac{x^{n}}{n!} + \dots$	$(\| x \| < \infty)$
$\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \dots + (- 1)^{n + 1} \frac{x^{n}}{n} + \dots$	$(- 1 < x \leq 1)$
$\ln (1 - x) = - (x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + \frac{x^{4}}{4} + \dots + \frac{x^{n}}{n} + \dots)$	$(- 1 \leq x < 1)$
$\sinh x = x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \frac{x^{7}}{7!} + \dots + \frac{x^{2 n + 1}}{(2 n + 1)!} + \dots$	$(\| x \| < \infty)$
$\cosh x = 1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \frac{x^{6}}{6!} + \dots + \frac{x^{2 n}}{(2 n)!} + \dots$	$(\| x \| < \infty)$