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! colspan="2" style="background: #eee;" | Series in symbolic forms | ! colspan="2" style="background: #eee;" | Series in symbolic forms | ||
|- | |- | ||
− | |<math> \ | + | |<math> \text{Taylor Series in one variable } = \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math> |
|- | |- | ||
− | |<math> \ | + | |<math> \text{Taylor Series in} d \text{ 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).\!</math> | \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).\!</math> | ||
|- | |- | ||
! colspan="2" style="background: #eee;" | Taylor Series of certain functions | ! colspan="2" style="background: #eee;" | Taylor Series of certain functions | ||
|- | |- | ||
− | |<math> \ | + | |<math> \text{Exponential } e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, \text{ for all } x\in {\mathbb C}\ </math> |
|- | |- | ||
− | |<math> \ | + | |<math> \text{Logarithm } ln(1+x) = \sum^{\infin}_{n=1} (-1)^{n+1}\frac{x^n}n,\text{ when }-1<x\le1</math> |
|- | |- | ||
! colspan="2" style="background: #eee;" | Geometric Series and related series | ! colspan="2" style="background: #eee;" | Geometric Series and related series | ||
|- | |- | ||
− | | [[more_on_geometric_series|(info)]] <math class="inline"> \ | + | | [[more_on_geometric_series|(info)]] <math class="inline"> \text{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. </math> |
|- | |- | ||
− | | [[more_on_geometric_series|(info)]] <math class="inline"> \ | + | | [[more_on_geometric_series|(info)]] <math class="inline"> \text{Infinite Geometric Series Formula } \sum_{k=0}^\infty x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. </math> |
|- | |- | ||
|<math>\frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n\quad\mbox{ for }|x| < 1 \text{ and } m\in\mathbb{N}_0\!</math> | |<math>\frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n\quad\mbox{ for }|x| < 1 \text{ and } m\in\mathbb{N}_0\!</math> | ||
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</math> | </math> | ||
|- | |- | ||
− | | Some particular Cases: | + | | <math> \text{Some particular Cases: } </math> |
|- | |- | ||
| <math> (a+x)^2 \ = \ a^2 \ + \ 2ax \ + \ x^2</math> | | <math> (a+x)^2 \ = \ a^2 \ + \ 2ax \ + \ x^2</math> | ||
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| <math> (a+x)^4 \ = \ a^4 \ + \ 4a^3x \ + \ 6a^2x^2 \ + \ 4ax^3 \ + \ x^4</math> | | <math> (a+x)^4 \ = \ a^4 \ + \ 4a^3x \ + \ 6a^2x^2 \ + \ 4ax^3 \ + \ x^4</math> | ||
|- | |- | ||
− | | <math> (a+x)^{-1} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \ | + | | <math> (a+x)^{-1} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdots </math> |
|<math> -1 < x < 1 \qquad </math> | |<math> -1 < x < 1 \qquad </math> | ||
|- | |- | ||
− | | <math> (a+x)^{-2} \ = \ 1 \ - \ 2x \ + \ 3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \ | + | | <math> (a+x)^{-2} \ = \ 1 \ - \ 2x \ + \ 3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \cdots </math> |
|<math> -1 < x < 1 \qquad</math> | |<math> -1 < x < 1 \qquad</math> | ||
|- | |- | ||
− | | <math> (a+x)^{-3} \ = \ 1 \ - \ 3x \ + \ 6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \ | + | | <math> (a+x)^{-3} \ = \ 1 \ - \ 3x \ + \ 6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \cdots </math> |
|<math> -1 < x < 1 \qquad</math> | |<math> -1 < x < 1 \qquad</math> | ||
|- | |- | ||
− | | <math> (a+x)^{-1/2} \ = \ 1 \ - \ \frac{1}{2}x \ + \ \frac{1 \ | + | | <math> (a+x)^{-1/2} \ = \ 1 \ - \ \frac{1}{2}x \ + \ \frac{1 \cdot 3}{2 \cdot 4}x^2 \ - \ \frac {1 \cdot 3 \cdot 5 }{2 \cdot 4 \cdot 6} x^3 \ + \ \cdots </math> |
| <math> -1 < x \leqq 1 \qquad </math> | | <math> -1 < x \leqq 1 \qquad </math> | ||
|- | |- | ||
− | | <math> (a+x)^{1/2} \ = \ 1 \ + \ \frac{1}{2}x \ - \ \frac{1 }{2 \ | + | | <math> (a+x)^{1/2} \ = \ 1 \ + \ \frac{1}{2}x \ - \ \frac{1 }{2 \cdot\ 4}x^2 \ + \ \frac {1 \cdot 3}{2 \cdot 4 \cdot 6} x^3 \ - \ \cdots </math> |
| <math> -1 < x \leqq 1 \qquad </math> | | <math> -1 < x \leqq 1 \qquad </math> | ||
|- | |- | ||
− | | <math> (a+x)^{-1/3} \ = \ 1 \ - \ \frac{1}{3}x \ + \ \frac{1 \ | + | | <math> (a+x)^{-1/3} \ = \ 1 \ - \ \frac{1}{3}x \ + \ \frac{1 \cdot 4}{3 \cdot 6}x^2 \ - \ \frac {1 \cdot 4 \cdot 7 }{3 \cdot 6 \cdot 9} x^3 \ + \ \cdots </math> |
| <math> -1 < x \leqq 1 \qquad </math> | | <math> -1 < x \leqq 1 \qquad </math> | ||
|- | |- | ||
− | | <math> (a+x)^{1/3} \ = \ 1 \ + \ \frac{1}{3}x \ - \ \frac{2}{3 \ | + | | <math> (a+x)^{1/3} \ = \ 1 \ + \ \frac{1}{3}x \ - \ \frac{2}{3 \cdot 6}x^2 \ + \ \frac {2 \cdot 5 }{3 \cdot 6 \cdot 9} x^3 \ - \ \cdots </math> |
| <math> -1 < x \leqq 1 \qquad </math> | | <math> -1 < x \leqq 1 \qquad </math> | ||
|- | |- | ||
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! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Series Expansion of Circular functions | ! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Series Expansion of Circular functions | ||
|- | |- | ||
− | | <math> \sin x \ = \ x \ - \ \frac{x^3}{3!} \ + \ \frac{x^5}{5!} \ - \ \frac{x^7}{7!} \ + \ \cdots | + | | <math> \sin x \ = \ x \ - \ \frac{x^3}{3!} \ + \ \frac{x^5}{5!} \ - \ \frac{x^7}{7!} \ + \ \cdots</math> |
| <math> - \infty < x < \infty \qquad </math> | | <math> - \infty < x < \infty \qquad </math> | ||
|- | |- | ||
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| <math> \left \vert x \right \vert < \frac {\pi}{2} \qquad </math> | | <math> \left \vert x \right \vert < \frac {\pi}{2} \qquad </math> | ||
|- | |- | ||
− | | <math> \frac{1}{\sin x} \ = \ \frac{1}{x} \ + \ \frac {x}{6} \ + \ \frac{7x^3}{360} \ + \ \frac{31x^5}{ | + | | <math> \frac{1}{\sin x} \ = \ \frac{1}{x} \ + \ \frac {x}{6} \ + \ \frac{7x^3}{360} \ + \ \frac{31x^5}{15120} \ + \ \cdots \ + \ \frac{2(2^{2n-1}-1)B_n x^{2n-1}}{(2n)!} \ + \ \cdots </math> |
| <math> 0 < \left \vert x \right \vert < \pi \qquad </math> | | <math> 0 < \left \vert x \right \vert < \pi \qquad </math> | ||
|- | |- | ||
− | |<math> \arcsin x = x + {1 \over 2}{x^3 \over 3} + \frac{1 \ | + | |<math> \arcsin x = x + {1 \over 2}{x^3 \over 3} + \frac{1 \cdot 3}{ 2 \cdot 4} {x^5 \over 5} + \frac {1 \cdot 3 \cdot 5}{ 2 \cdot 4 \cdot 6}{x^7 \over 7} + \cdots </math> |
|<math> \left \vert x \right \vert < 1 \qquad</math> | |<math> \left \vert x \right \vert < 1 \qquad</math> | ||
|- | |- | ||
− | |<math> \arccos x = {\pi \over 2} - \sin ^{-1} x = {\pi \over 2} - \left ( x + {1 \over 2}{x^3 \over 3} +\frac{1 \ | + | |<math> \arccos x = {\pi \over 2} - \sin ^{-1} x = {\pi \over 2} - \left ( x + {1 \over 2}{x^3 \over 3} +\frac{1 \cdot 3}{2 \cdot 4} {x^5 \over 5} + \cdots \ \right )</math> |
|<math>\left \vert x \right \vert < 1 \qquad</math> | |<math>\left \vert x \right \vert < 1 \qquad</math> | ||
|- | |- | ||
|<math> \arctan x = | |<math> \arctan x = | ||
\begin{cases} | \begin{cases} | ||
− | x - {x^3 \over 3} + {x^5 \over 5} - { x^7 \over 7} + \cdots & | + | x - {x^3 \over 3} + {x^5 \over 5} - { x^7 \over 7} + \cdots, & \left \vert x \right \vert < 1 \\ |
− | + | {\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\mbox{ if } x \geqq 1 \\ | |
+ | -{\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\mbox{ if } x \leqq -1 | ||
\end{cases} | \end{cases} | ||
</math> | </math> | ||
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|<math> \arccot x = {\pi \over 2} - \arctan x = | |<math> \arccot x = {\pi \over 2} - \arctan x = | ||
\begin{cases} | \begin{cases} | ||
− | {\pi \over 2} - \left ( x - {x^3 \over 3} + {x^5 \over 5} - \cdots \right ) & | + | {\pi \over 2} - \left ( x - {x^3 \over 3} + {x^5 \over 5} - \cdots \right ), &\left \vert x \right \vert < 1 \\ |
− | + | {\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x > 1\\ | |
+ | -{\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x < -1 | ||
\end{cases} | \end{cases} | ||
</math> | </math> | ||
|- | |- | ||
− | |<math> \arccos ({1 \over x}) = {\pi \over 2} - \left ( {1 \over x} + \frac{1}{2 \ | + | |<math> \arccos ({1 \over x}) = {\pi \over 2} - \left ( {1 \over x} + \frac{1}{2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots \right )</math> |
|<math>\left \vert x \right \vert > 1 \qquad</math> | |<math>\left \vert x \right \vert > 1 \qquad</math> | ||
|- | |- | ||
− | |<math> \arcsin ({1 \over x}) = {1 \over x} + {1 \over 2 \ | + | |<math> \arcsin ({1 \over x}) = {1 \over x} + {1 \over 2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots</math> |
− | |<math> \left \vert x \right \vert > 1 | + | |<math> \left \vert x \right \vert > 1</math> |
|- | |- | ||
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Series Expansion of Hyperbolic functions | ! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Series Expansion of Hyperbolic functions | ||
|- | |- | ||
− | | <math>\, \ | + | | <math>\, \sinh x = x + {x^3 \over 3!} + {x^5 \over 5!} + { x^7 \over 7!} + \cdots\,</math> |
| <math> - \infty < x < \infty \qquad</math> | | <math> - \infty < x < \infty \qquad</math> | ||
|- | |- | ||
− | | <math>\, \ | + | | <math>\, \cosh x = 1 + {x^2 \over 2!} + {x^4 \over 4!} + { x^6 \over 6!} + \cdots\,</math> |
| <math> - \infty < x < \infty \qquad</math> | | <math> - \infty < x < \infty \qquad</math> | ||
|- | |- | ||
− | | <math>\, \ | + | | <math>\, \tanh 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\,</math> |
| <math> \vert x \vert < {\pi \over 2} \qquad</math> | | <math> \vert x \vert < {\pi \over 2} \qquad</math> | ||
|- | |- | ||
− | | <math>\, \coth | + | | <math>\, \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\,</math> |
| <math> 0 < \vert x \vert < \pi \qquad</math> | | <math> 0 < \vert x \vert < \pi \qquad</math> | ||
|- | |- | ||
− | |<math>\frac {1}{\ | + | |<math>\frac {1}{\cosh x} = 1 - {x2 \over 2} + {5x^4 \over 24} -{61x^6 \over 720} + \cdots \frac{(-1)^nE_nx^{2n}}{(2n)!} + \cdots</math> |
|<math>\vert x \vert < {\pi \over 2}</math> | |<math>\vert x \vert < {\pi \over 2}</math> | ||
|- | |- | ||
− | |<math> \frac{1}{\ | + | |<math> \frac{1}{\sinh 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</math> |
|<math>0 < \vert x \vert < \pi </math> | |<math>0 < \vert x \vert < \pi </math> | ||
|- | |- | ||
|- | |- | ||
− | |<math> \operatorname{ | + | |<math> \operatorname{arsinh}\,x = |
\begin{cases} | \begin{cases} | ||
− | x - {x^3 \over 2 \ | + | x - {x^3 \over 2 \cdot 3} + {1 \cdot 3 x^5 \cdot 2 \cdot 4 \cdot 5} - {1 \cdot 3 \cdot 5 x^7 \over 2 \cdot 4 \cdot 6 \cdot 7} + \cdots, & \left \vert x \right \vert < 1 \\ |
− | + | \left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \geqq 1\\ | |
− | \ | + | -\left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \leqq -1 |
− | + | ||
− | + | ||
− | \ | + | |
− | \ \right | + | |
\end{cases} | \end{cases} | ||
</math> | </math> | ||
|- | |- | ||
− | |<math> \operatorname{ | + | |<math> \operatorname{arcosh} \,x = \begin{cases} |
− | + | \{ \ln (2x) - ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh}\,x > 0, x \geqq 1 \\ | |
− | + | - \{ \ln (2x) - ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh} \,x < 0, x \geqq 1 | |
− | + | \end{cases} </math> | |
− | + | ||
− | \end{ | + | |
− | + | ||
− | + | ||
− | </math> | + | |
|- | |- | ||
|<math> \operatorname{argth} \,x = x + { x^3 \over 5} + {x^5 \over 5 } + {x^7 \over 7 }+ \cdots </math> | |<math> \operatorname{argth} \,x = x + { x^3 \over 5} + {x^5 \over 5 } + {x^7 \over 7 }+ \cdots </math> | ||
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! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Series of Reciprocal Power Series | ! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Series of Reciprocal Power Series | ||
|- | |- | ||
− | + | | <math>\text{if }\ y = c_1x +c_2x^3 +c_3x^3 + c_4x^4 + c_5x^5 + c_6x^6 + \cdots\,\qquad \text{then }\ x = C_1y+C_2y^2+C_3y^3+C_4y^4+C_5y^5+C_6y^6+\cdots </math> | |
− | + | ||
− | | <math>\ | + | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
|- | |- | ||
− | |<math>c_1^ | + | |<math>\text{where }\ c_1C_1 = 1, \qquad c_1^3C_2= -c_2, \qquad c_1^7C_3 = 2c_2^2 - c_1c_3 </math> |
|- | |- | ||
− | |<math>c_1^9C_5 = 6c_1^2c_2c_4 + </math> | + | |<math>c_1^7C_4 = 5c_1c_2c_3 - 5c_2^3 - c_2^2c_4, \qquad c_1^9C_5 = 6c_1^2c_2c_4 + </math> |
|- | |- | ||
|<math>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 </math> | |<math>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 </math> |
Revision as of 17:43, 12 November 2011
If you enjoy using this collective table of formulas, please consider donating to Project Rhea or becoming a sponsor. |
Power Series Formulas | ||
---|---|---|
Series in symbolic forms | ||
$ \text{Taylor Series in one variable } = \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} $ | ||
$ \text{Taylor Series in} d \text{ 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 | ||
$ \text{Exponential } e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, \text{ for all } x\in {\mathbb C}\ $ | ||
$ \text{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) $ \text{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) $ \text{Infinite Geometric Series Formula } \sum_{k=0}^\infty 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\! $ | ||
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} $ | ||
$ \text{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 \ - \ \cdots $ | $ -1 < x < 1 \qquad $ | |
$ (a+x)^{-2} \ = \ 1 \ - \ 2x \ + \ 3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \cdots $ | $ -1 < x < 1 \qquad $ | |
$ (a+x)^{-3} \ = \ 1 \ - \ 3x \ + \ 6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \cdots $ | $ -1 < x < 1 \qquad $ | |
$ (a+x)^{-1/2} \ = \ 1 \ - \ \frac{1}{2}x \ + \ \frac{1 \cdot 3}{2 \cdot 4}x^2 \ - \ \frac {1 \cdot 3 \cdot 5 }{2 \cdot 4 \cdot 6} x^3 \ + \ \cdots $ | $ -1 < x \leqq 1 \qquad $ | |
$ (a+x)^{1/2} \ = \ 1 \ + \ \frac{1}{2}x \ - \ \frac{1 }{2 \cdot\ 4}x^2 \ + \ \frac {1 \cdot 3}{2 \cdot 4 \cdot 6} x^3 \ - \ \cdots $ | $ -1 < x \leqq 1 \qquad $ | |
$ (a+x)^{-1/3} \ = \ 1 \ - \ \frac{1}{3}x \ + \ \frac{1 \cdot 4}{3 \cdot 6}x^2 \ - \ \frac {1 \cdot 4 \cdot 7 }{3 \cdot 6 \cdot 9} x^3 \ + \ \cdots $ | $ -1 < x \leqq 1 \qquad $ | |
$ (a+x)^{1/3} \ = \ 1 \ + \ \frac{1}{3}x \ - \ \frac{2}{3 \cdot 6}x^2 \ + \ \frac {2 \cdot 5 }{3 \cdot 6 \cdot 9} x^3 \ - \ \cdots $ | $ -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}{15120} \ + \ \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 \cdot 3}{ 2 \cdot 4} {x^5 \over 5} + \frac {1 \cdot 3 \cdot 5}{ 2 \cdot 4 \cdot 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 \cdot 3}{2 \cdot 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, & \left \vert x \right \vert < 1 \\ {\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\mbox{ if } x \geqq 1 \\ -{\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\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 ), &\left \vert x \right \vert < 1 \\ {\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x > 1\\ -{\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x < -1 \end{cases} $ | ||
$ \arccos ({1 \over x}) = {\pi \over 2} - \left ( {1 \over x} + \frac{1}{2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots \right ) $ | $ \left \vert x \right \vert > 1 \qquad $ | |
$ \arcsin ({1 \over x}) = {1 \over x} + {1 \over 2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots $ | $ \left \vert x \right \vert > 1 $ | |
Series Expansion of Hyperbolic functions | ||
$ \, \sinh x = x + {x^3 \over 3!} + {x^5 \over 5!} + { x^7 \over 7!} + \cdots\, $ | $ - \infty < x < \infty \qquad $ | |
$ \, \cosh x = 1 + {x^2 \over 2!} + {x^4 \over 4!} + { x^6 \over 6!} + \cdots\, $ | $ - \infty < x < \infty \qquad $ | |
$ \, \tanh 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}{\cosh 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}{\sinh 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{arsinh}\,x = \begin{cases} x - {x^3 \over 2 \cdot 3} + {1 \cdot 3 x^5 \cdot 2 \cdot 4 \cdot 5} - {1 \cdot 3 \cdot 5 x^7 \over 2 \cdot 4 \cdot 6 \cdot 7} + \cdots, & \left \vert x \right \vert < 1 \\ \left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \geqq 1\\ -\left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \leqq -1 \end{cases} $ | ||
$ \operatorname{arcosh} \,x = \begin{cases} \{ \ln (2x) - ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh}\,x > 0, x \geqq 1 \\ - \{ \ln (2x) - ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh} \,x < 0, x \geqq 1 \end{cases} $ | ||
$ \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 | ||
$ \text{if }\ y = c_1x +c_2x^3 +c_3x^3 + c_4x^4 + c_5x^5 + c_6x^6 + \cdots\,\qquad \text{then }\ x = C_1y+C_2y^2+C_3y^3+C_4y^4+C_5y^5+C_6y^6+\cdots $ | ||
$ \text{where }\ c_1C_1 = 1, \qquad c_1^3C_2= -c_2, \qquad 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, \qquad 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 $ |