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Table of Taylor Series
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

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