| 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 | ||
| $ 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 & \left \vert x \right \vert < 1 \\ \pm {\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5 + \cdots & [ + \mbox{ if } x \geqq 1 , - \mbox{ if } x leqq -1 \ ] \end{cases} $ | 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)\, $ | |
