| Power Series Formulas | ||||
|---|---|---|---|---|
| Series in symbolic forms | ||||
| Taylor Series in one variable | $ \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} $ | |||
| Taylor Series in d variables |
$ =\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin} \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d).\! $ | |||
| Taylor Series of certain functions | ||||
| exponential | $ e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, $ $ \text{ for all } x\in {\mathbb C}\ $ | |||
| logarithm |
$ \ln(1+x) = \sum^{\infin}_{n=1} (-1)^{n+1}\frac{x^n}n,\text{ when }-1<x\le1 $ | |||
| Geometric Series and related series | ||||
| (info) Finite Geometric Series Formula | $ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right. $ | |||
| (info) Infinite Geometric Series Formula | $ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. $ | |||
| $ \frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n\quad\mbox{ for }|x| < 1 \text{ and } m\in\mathbb{N}_0\! $ | ||||
|
$ \frac{x}{(1-x)^2} = \sum^{\infin}_{n=1}n x^n\quad\text{ for }|x| < 1\! $ | ||||
| Other Series | ||||
| notes/name | equation | |||
| Taylor series of Single Variable Functions | ||||
| $ \,f(x) \ = \ f(a) \ + \ f'(a)(x \ - \ a) \ + \ \frac{f''(a)(x-a)^2}{2!} \ + \ \cdot \cdot \cdot \ + \ \frac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!} \ + \ R_n \, $ | ||||
| $ \text{Rest of Lagrange } \qquad R_n = \frac {f^{(n)}(\zeta)(x-a)^n}{n!} $ | ||||
| $ \text{Rest of Cauchy } \qquad R_n = \frac {f^{(n)}(\zeta)(x-\zeta)^{n-1}(x-a)}{(n-1)!} $ | ||||
| Binomial Series | ||||
| $ \begin{align} (a+x)^n & = a^n + na^{n-1}x + \frac {n(n-1)}{2!} a^{n-2}x^2 + \frac {n(n-1)(n-2)}{3!} a^{n-3}x^3 + \cdot \cdot \cdot \\ & = a^n + \binom{n}{1} a^{n-1}x + \binom{n}{2} a^{n-2}x^2 + \binom{n}{3} a^{n-3}x^3 + \cdot \cdot \cdot \\ \end{align} $ | ||||
| Some particular Cases: | ||||
| $ (a+x)^2 \ = \ a^2 \ + \ 2ax \ + \ x^2 $ | ||||
| $ (a+x)^3 \ = \ a^3 \ + \ 3a^2x \ + \ 3ax^2 \ + \ x^3 $ | ||||
| $ (a+x)^4 \ = \ a^4 \ + \ 4a^3x \ + \ 6a^2x^2 \ + \ 4ax^3 \ + \ x^4 $ | ||||
| $ (a+x)^{-1} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdot \cdot \cdot $ | $ -1 < x < 1 \qquad $ | |||
| $ (a+x)^{-2} \ = \ 1 \ - \ 2x \ + \ 3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \cdot \cdot \cdot $ | $ -1 < x < 1 \qquad $ | |||
| $ (a+x)^{-3} \ = \ 1 \ - \ 3x \ + \ 6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \cdot \cdot \cdot $ | $ -1 < x < 1 \qquad $ | |||
| $ (a+x)^{-1/2} \ = \ 1 \ - \ \frac{1}{2}x \ + \ \frac{1 \bullet 3}{2 \bullet 4}x^2 \ - \ \frac {1 \bullet 3 \bullet 5 }{2 \bullet 4 \bullet 6} x^3 \ + \ \cdot \cdot \cdot $ | $ -1 < x \leqq 1 \qquad $ | |||
| $ (a+x)^{1/2} \ = \ 1 \ + \ \frac{1}{2}x \ - \ \frac{1 }{2 \bullet 4}x^2 \ + \ \frac {1 \bullet 3 }{2 \bullet 4 \bullet 6} x^3 \ - \ \cdot \cdot \cdot $ | $ -1 < x \leqq 1 \qquad $ | |||
| $ (a+x)^{-1/3} \ = \ 1 \ - \ \frac{1}{3}x \ + \ \frac{1 \bullet 4}{3 \bullet 6}x^2 \ - \ \frac {1 \bullet 4 \bullet 7 }{3 \bullet 6 \bullet 9} x^3 \ + \ \cdot \cdot \cdot $ | $ -1 < x \leqq 1 \qquad $ | |||
| $ (a+x)^{1/3} \ = \ 1 \ + \ \frac{1}{3}x \ - \ \frac{2}{3 \bullet 6}x^2 \ + \ \frac {2 \bullet 5 }{3 \bullet 6 \bullet 9} x^3 \ - \ \cdot \cdot \cdot $ | $ -1 < x \leqq 1 \qquad $ | |||
| Series Expansion of Exponential functions and Logarithms | ||||
| $ e^x \ = \ 1 \ + \ x \ + \ \frac{x^2}{2!} \ + \ \frac{x^3}{3!} \ + \ \cdots $ | $ - \infty < x < \infty \qquad $ | |||
| $ a^x \ = \ e^{x \ln a} \ = \ 1 \ + \ x \ln a \ + \ \frac{(x \ln a)^2}{2!} \ + \ \frac{(x \ln a)^3}{3!} \ + \ \cdots $ | $ - \infty < x < \infty \qquad $ | |||
| $ \ln(1+x) \ = \ x \ - \ \frac{x^2}{2} \ + \ \frac{x^3}{3} \ - \ \frac{x^4}{4} \ + \ \cdots $ | $ -1 < x \leqq 1 \qquad $ | |||
| $ \frac{1}{2} \ln \left ( \frac {1+x}{1-x} \right ) \ = \ x \ + \ \frac{x^3}{3} \ + \ \frac {x^5}{5} \ + \ \frac{x^7}{7} \ + \ \cdots \ $ | $ -1 < x < 1 \qquad $ | |||
| $ \ln x \ = \ 2 \left \{ \left ( \frac {x-1}{x+1} \right ) \ + \ \frac{1}{3} \left ( \frac {x-1}{x+1} \right ) ^3 \ + \ \frac{1}{5} \left ( \frac{x-1}{x+1} \right ) ^ 5 \ + \ \cdots \ \right \} $ | $ x > 0 \qquad $ | |||
| $ \ln x \ = \ \left ( \frac {x-1}{x} \right ) \ + \ \frac{1}{2} \left ( \frac {x-1}{x} \right ) ^2 \ + \ \frac{1}{3} \left ( \frac{x-1}{x} \right ) ^ 3 \ + \ \cdots \ $ | $ x \geqq \frac {1}{2} \qquad $ | |||
| Series Expansion of Circular functions | ||||
| $ \sin x \ = \ x \ - \ \frac{x^3}{3!} \ + \ \frac{x^5}{5!} \ - \ \frac{x^7}{7!} \ + \ \cdots \ $ | $ - \infty < x < \infty \qquad $ | |||
| $ \cos x \ = \ 1 \ - \ \frac{x^2}{2!} \ + \ \frac{x^4}{4!} \ - \ \frac{x6}{6!} \ + \ \cdots $ | $ - \infty < x < \infty \qquad $ | |||
| $ \cot x \ = \ \frac{1}{x} \ - \ \frac {x}{3} \ - \ \frac{x^3}{45} \ - \ \frac{2x^5}{945} \ - \ \cdots \ - \ \frac{2^{2n}B_n x^{2n-1}}{(2n)!} \ - \ \cdots $ | $ 0 < \left \vert x \right \vert < \pi \qquad $ | |||
| $ \frac{1}{\cos x} \ = \ 1 \ + \ \frac {x^2}{2} \ + \ \frac{x^4}{24} \ + \ \frac{61x^6}{720} \ + \ \cdots \ - \ \frac{E_n x^{2n}}{(2n)!} \ + \ \cdots $ | $ \left \vert x \right \vert < \frac {\pi}{2} \qquad $ | |||
| $ \frac{1}{\sin x} \ = \ \frac{1}{x} \ + \ \frac {x}{6} \ + \ \frac{7x^3}{360} \ + \ \frac{31x^5}{15,120} \ + \ \cdots \ + \ \frac{2(2^{2n-1}-1)B_n x^{2n-1}}{(2n)!} \ + \ \cdots $ | $ 0 < \left \vert x \right \vert < \pi \qquad $ | |||
| $ \arcsin x = x + {1 \over 2}{x^3 \over 3} + \frac{1 \bullet 3}{ 2 \bullet 4} {x^5 \over 5} + \frac {1 \bullet 3 \bullet 5}{ 2 \bullet 4 \bullet 6}{x^7 \over 7} + \cdots $ | $ \left \vert x \right \vert < 1 \qquad $ | |||
| $ \arccos x = {\pi \over 2} - \sin ^{-1} x = {\pi \over 2} - \left ( x + {1 \over 2}{x^3 \over 3} +\frac{1 \bullet 3}{2 \bullet 4} {x^5 \over 5} + \cdots \ \right ) $ | $ \left \vert x \right \vert < 1 \qquad $ | |||
| $ \arctan x = \begin{cases} x - {x^3 \over 3} + {x^5 \over 5} - { x^7 \over 7} + \cdots & \qquad \left \vert x \right \vert < 1 \\ \pm {\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots & \qquad [ + \mbox{ if } x \geqq 1 , - \mbox{ if } x \leqq -1 \ ] \\ \end{cases} $ | ||||
| $ \arccot x = {\pi \over 2} - \arctan x = \begin{cases} {\pi \over 2} - \left ( x - {x^3 \over 3} + {x^5 \over 5} - \cdots \right ) & \qquad \qquad \qquad \qquad \left \vert x \right \vert < 1 \\ p {\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots & \qquad \qquad [ p = 0 \mbox{ if } x > 1 , p=1 \mbox{ if } x < -1 \ ] \\ \end{cases} $ | ||||
| $ \arccos ({1 \over x}) = {\pi \over 2} - \left ( {1 \over x} + \frac{1}{2 \bullet 3 x^3} + \frac{1 \bullet 3}{2 \bullet 4 \bullet 5 x^5} + \cdots \right ) $ | $ \left \vert x \right \vert > 1 \qquad $ | |||
| $ \arcsin ({1 \over x}) = {1 \over x} + {1 \over 2 \bullet 3 x^3} + \frac{1 \bullet 3}{2 \bullet 4 \bullet 5 x^5} + \cdots $ | $ \left \vert x \right \vert > 1 \qquad $ | |||
| Series Expansion of Hyperbolic functions | ||||
| $ \, \operatorname{sh}\, x = x + {x^3 \over 3!} + {x^5 \over 5!} + { x^7 \over 7!} + \cdots\, $ | $ - \infty < x < \infty \qquad $ | |||
| $ \, \operatorname{ch}\, x = 1 + {x^2 \over 2!} + {x^4 \over 4!} + { x^6 \over 6!} + \cdots\, $ | $ - \infty < x < \infty \qquad $ | |||
| $ \, \operatorname{th}\, x = x - {x^3 \over 3} + {2x^5 \over 15} - { 17x^7 \over 315} + \cdots \ \frac{(-1)^{n-1}2^{2n}(2^{2n} -1)B_nx^{2n-1}}{(2n)!} + \cdots\, $ | $ \vert x \vert < {\pi \over 2} \qquad $ | |||
| $ \, \coth \, x = {1 \over x} + {x \over 3} - {x^3 \over 45} + { 2x^5 \over 945} + \cdots \frac{(-1)^{n-1}2^{2n}b_nx^{2n-1}}{(2n)!} + \cdots\, $ | $ 0 < \vert x \vert < \pi \qquad $ | |||
| $ \frac {1}{\operatorname{ch}\, x} = 1 - {x2 \over 2} + {5x^4 \over 24} -{61x^6 \over 720} + \cdots \frac{(-1)^nE_nx^{2n}}{(2n)!} + \cdots $ | $ \vert x \vert < {\pi \over 2} $ | |||
| $ \frac{1}{\operatorname{sh}\, x} = {1 \over x} - {x \over 6} + {7x^3 \over 360} - {31x^5 \over 15,120} + \cdots \frac{(-1)^n2(2^{2n-1}-1)B_nx^{2n-1}}{(2n)!} + \cdots $ | $ 0 < \vert x \vert < \pi $ | |||
| $ \operatorname{argsh}\,x = \begin{cases} x - {x^3 \over 2 \bullet 3} + {1 \bullet 3 x^5 \over 2 \bullet 4 \bullet 5} - {1 \bullet 3 \bullet 5 x^7 \over 2 \bullet 4 \bullet 6 \bullet 7} + \cdots & \qquad \qquad \qquad \qquad \left \vert x \right \vert < 1 \\ \pm \left ( \ln \vert 2x \vert + {1 \over 2 \bullet 2 x^2} - {1 \bullet 3 \over 2 \bullet 4 \bullet 4x^4} + {1 \bullet 3 \bullet 5 \over 2 \bullet 4 \bullet 6 \bullet 6x^6} - \cdots \right )& \qquad \qquad \left [ \begin{matrix} + \ if \ x \geqq 1 \\ - \ if \ x \leqq -1 \end{matrix} \ \right ] \end{cases} $ | ||||
| $ \operatorname{argch} \,x = \pm \left \{ \ln (2x) - \left ( \frac{1}{2 \bullet 2x^2} + \frac{1 \bullet 3}{2 \bullet 4 \bullet 4x^4} + \frac { 1 \bullet 3 \bullet 5}{2 \bullet 4 \bullet 6 \bullet 6x^6} + \cdots \right ) \right \} \left [ \begin{matrix} + \ if \ \operatorname{argsh}\,x > 0, x \geqq 1 \\ - \ if \ \operatorname{argsh}\,x < 0, x \geqq 1 \end{matrix} \ \right ] $ | ||||
| $ \operatorname{argth} \,x = x + { x^3 \over 5} + {x^5 \over 5 } + {x^7 \over 7 }+ \cdots $ | $ \vert x \vert < 1 \qquad $ | |||
| $ \operatorname{argcoth} \,x = {1 \over x} + { 1 \over 3x^3} + {1 \over 5x^5 } + {1 \over 7x^7 }+ \cdots $ | $ \vert x \vert > 1 \qquad $ | |||
| Various Series | ||||
| $ \, e^{\sin x} = 1 + x + {x^2 \over 2} - {x^4 \over 8} - {x^5 \over 15} + \cdots\, $ | $ - \infty < x < \infty $ | |||
| $ \, e^{\cos x} = e \left ( 1 - {x^2 \over 2} + {x^4 \over 6} - {31x^6 \over 720} + \cdots \right ) \, $ | $ - \infty < x < \infty $ | |||
| $ \, e^{\tan x} = 1 + x + {x^2 \over 2} + {x^3 \over 2} + {3x^4 \over 8} + \cdots \, $ | $ \vert x \vert < { \pi \over 2} $ | |||
| $ e^x \sin x = x + x^2 + {2x^3 \over 3 } - {x^5 \over 30} - {x^6 \over 90} + \cdots + \frac{2^{n/2} \sin (n \pi /4)\ x^n}{n!} + \cdots $ | $ - \infty < x < \infty $ | |||
| $ e^x \cos x = 1 + x - {x^3 \over 3 } - {x^4 \over 6} + \cdots + \frac{2^{n/2} \cos (n \pi /4)\ x^n}{n!} + \cdots $ | $ - \infty < x < \infty $ | |||
| $ \ln \vert \sin x \vert = \ln \vert x \vert - {x^2 \over 6} - {x^4 \over 180} - {x^6 \over 2835} - \cdots - \frac{2^{2n-1}B_nx^{2n}}{n(2n)!} + \cdots $ | $ 0 < \vert x \vert < \pi $ | |||
| $ \ln \vert \cos x \vert = - {x^2 \over 2} - {x^4 \over 12} - {x^6 \over 45} - {17x^8 \over 2520} - \cdots - \frac{2^{2n-1}(2^{2n}-1)B_nx^{2n}}{n(2n)!} + \cdots $ | $ \vert x \vert < {\pi \over 2} $ | |||
| $ \ln \vert \tan x \vert = \ln \vert x \vert + {x^2 \over 3} + {7x^4 \over 90} + {62x^6 \over 2835}+ \cdots + \frac{2^{2n}(2^{2n-1}-1)B_nx^{2n}}{n(2n)!} + \cdots $ | $ 0 < \vert x \vert < {\pi \over 2} $ | |||
| $ \frac{\ln (1+x)}{1+x} = x - (1+ {1 \over 2})^{x^2} + (1 + {1 \over 2} + {1 \over 3})^{x^3} - \cdots $ | $ \vert x \vert < 1 $ | Series of Reciprocal Power Series | ||
| if | ||||
| $ \, y = c_1x +c_2x^3 +c_3x^3 + c_4x^4 + c_5x^5 + c_6x^6 + \cdots\, $ | ||||
| then | ||||
| $ x = C_1y+C_2y^2+C_3y^3+C_4y^4+C_5y^5+C_6y^6+ $ | ||||
| $ c_1C_1 = 1 $ | ||||
| $ c_1^3C_2= -c_2 $ | ||||
| $ c_1^7C_3 = 2c_2^2 - c_1c_3 $ | ||||
| $ c_1^7C_4 = 5c_1c_2c_3 - 5c_2^3 - c_2^2c_4 $ | ||||
| $ c_1^9C_5 = 6c_1^2c_2c_4 + $ | ||||
| $ c_1^{11}C_6 = 7 c_1^3c_2 c_5 + 84 c_1 c_2^3c_3 + 7c_1^3c_3c_4 - 28c_1^2c_2c_3^2 - c_1^4c/-6 - 28c_1^2c_2^2c_4 - 42c_2^5 $ | ||||
| Taylor Series of Two Variables function | ||||
| $ \, f(x,y) = f(a,b) + (x-a)f_x(a,b) + (y-b)f_y(a,b) + {1 \over 2!} \left \{ (x-a)^2f_{xx}(a,b) + 2(x-a)(y-b)f_{xy}(a,b)+(y-b)^2f_{yy}(a,b) \right \} + \cdots\, $ | ||||
| $ f_x(a,b),f_y(a,b) , \cdots \text {denote the partial derivatives with respect to} x , y \cdots $ | ||||
