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|<math> -1 < x < 1 \qquad</math> | |<math> -1 < x < 1 \qquad</math> | ||
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| − | | <math> (a+x)^{-1/2} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdot \cdot \cdot </math> | + | | <math> (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 </math> |
| − | | <math> -1 < x < 1 \qquad </math> | + | | <math> -1 < x \leqq 1 \qquad </math> |
| + | |- | ||
| + | | <math> (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 </math> | ||
| + | | <math> -1 < x \leqq 1 \qquad </math> | ||
| + | |- | ||
| + | | <math> (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 </math> | ||
| + | | <math> -1 < x \leqq 1 \qquad </math> | ||
| + | |- | ||
| + | | <math> (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 </math> | ||
| + | | <math> -1 < x \leqq 1 \qquad </math> | ||
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| − | |||
! style="background: rgb(238, 238, 238) none ! 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 Exponential functions and Logarithms | ! style="background: rgb(238, 238, 238) none ! 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 Exponential functions and Logarithms | ||
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Revision as of 16:35, 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 $ | $ -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 | |
| 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)\, $ |
