Power Series

ex=1+11!x+12!x2+13!x3+...e^x=1+\frac{1}{1!}x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+...
sin⁡(x)=11!x−13!x3+15!x5−...\sin (x) = \frac{1}{1!}x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-...
cos⁡(x)=1−12!x2+14!x4−...\cos (x) = 1 - \frac{1}{2!}x^2 + \frac{1}{4!}x^4 - ...
tan⁡(x)=x+13x3+115!x5+...∣x∣<π2\tan (x) = x + \frac{1}{3}x^3 + \frac{1}{15!}x^5 + ... |x|<\frac{\pi}{2}
ln⁡(1+x)=x+12x2+13x3+...∣x∣<1\ln (1+x) = x + \frac{1}{2}x^2 + \frac{1}{3}x^3+... |x|<1
(1+x)a=1+a1!x+a(a−1)2!x2+...(1+x)^a = 1 + \frac{a}{1!}x+\frac{a(a-1)}{2!}x^2 + ...
1+x=1+12x+18x2+116x3+...\sqrt{1+x} = 1 +\frac{1}{2}x + \frac{1}{8}x^2 + \frac{1}{16}x^3 + ...
11+x=1−12x+38x2−516x3+...\frac{1}{\sqrt{1+x}}=1 - \frac{1}{2}x+\frac{3}{8}x^2 - \frac{5}{16}x^3 + ...
arctan⁡(x)=x−13x3+15x5−...∣x∣<1\arctan (x) = x - \frac{1}{3}x^3 + \frac{1}{5}x^5 - ... |x|<1
arcsin⁡(x)=x+16x3+340x5+...∣x∣<1\arcsin (x) = x +\frac{1}{6}x^3 + \frac{3}{40}x^5 + ... |x|<1
cosh⁡(x)=1+12!x2+14!x4+...\cosh (x) = 1 + \frac{1}{2!}x^2 + \frac{1}{4!}x^4 + ...
sinh⁡(x)=x+13!x3+15!x5+...\sinh (x) = x + \frac{1}{3!}x^3 + \frac{1}{5!}x^5 + ...