Table of Taylor Series | |
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Taylor series of functions of one variable | |

$ 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 $ | |

Rn is the rest of the first n terms, and can be placed in one of two forms: | |

$ \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)!} $ | |

$ \text{z value, which may be different in two residues, located between a and x. the result is valid if f(x) has continuous derivatives at least up to order n } $ | |

$ \text {if } \lim_{n \to \infty}R_n \ = \ 0,\ \text{ the infinite series obtained is called the taylor series of f(x) near x = a } $ | |

$ \text { if a =0 , it is often called Mac Laurin series } $ | |

Binomial Series | |

Expansion Series of Exponential functions and logarithms | |

Expansion Series of Circular Functions | |

Expansion Series of Hyperbolic Functions | |

Various Series | |

series of reciprocal power series | |

Taylor Series of Functions of two variables | |