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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 \ - \ \cdot \cdot \cdot \qquad -1<x<1</math> | + | | <math> (a+x)^{-1} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdot \cdot \cdot \qquad </math> |
| + | |<math> -1<x<1 </math> | ||
|- | |- | ||
| − | | <math> (a+x)^{-2} \ = \ 1 \ - \ 2x \ + \ 3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \cdot \cdot \cdot \qquad - 1<x<1</math> | + | | <math> (a+x)^{-2} \ = \ 1 \ - \ 2x \ + \ 3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \cdot \cdot \cdot \qquad</math> |
| + | |<math> - 1<x<1 </math> | ||
|- | |- | ||
| − | | <math> (a+x)^{-3} \ = \ 1 \ - \ 3x \ + \ 6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \cdot \cdot \cdot \qquad -1<x<1</math> | + | | <math> (a+x)^{-3} \ = \ 1 \ - \ 3x \ + \ 6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \cdot \cdot \cdot \qquad </math> |
| + | |<math>-1<x<1</math> | ||
|- | |- | ||
| − | | <math> (a+x)^{-1/2} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdot \cdot \cdot \qquad -1<x<1</math> | + | | <math> (a+x)^{-1/2} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdot \cdot \cdot \qquad </math> |
| + | | <math> -1<x<1</math> | ||
|- | |- | ||
Revision as of 13:57, 22 November 2010
| Taylor Series | |
|---|---|
| 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 \qquad $ | $ -1<x<1 $ |
| $ (a+x)^{-2} \ = \ 1 \ - \ 2x \ + \ 3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \cdot \cdot \cdot \qquad $ | $ - 1<x<1 $ |
| $ (a+x)^{-3} \ = \ 1 \ - \ 3x \ + \ 6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \cdot \cdot \cdot \qquad $ | $ -1<x<1 $ |
| $ (a+x)^{-1/2} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdot \cdot \cdot \qquad $ | $ -1<x<1 $ |
| Series Expansion of Exponential functions and Logarithms | |
| The complement of an event A (i.e. the event A not occurring) | $ \,P(A^c) = 1 - P(A)\, $ |
| Series Expansion of Circular functions | |
| The complement of an event A (i.e. the event A not occurring) | $ \,P(A^c) = 1 - P(A)\, $ |
| Series Expansion of Hyperbolic functions | |
| The complement of an event A (i.e. the event A not occurring) | $ \,P(A^c) = 1 - P(A)\, $ |
| Various Series | |
| The complement of an event A (i.e. the event A not occurring) | $ \,P(A^c) = 1 - P(A)\, $ |
| Series of Reciprocal Power Series | |
| The complement of an event A (i.e. the event A not occurring) | $ \,P(A^c) = 1 - P(A)\, $ |
| Taylor Series of Two Variables function | |
| The complement of an event A (i.e. the event A not occurring) | $ \,P(A^c) = 1 - P(A)\, $ |
