# This Quantum World/Appendix/Taylor series

#### Taylor series[edit]

A well-behaved function can be expanded into a power series. This means that for all non-negative integers there are real numbers such that

Let us calculate the first four derivatives using :

Setting equal to zero, we obtain

Let us write for the -th derivative of We also write — think of as the "zeroth derivative" of We thus arrive at the general result where the *factorial* is defined as equal to 1 for and and as the product of all natural numbers for Expressing the coefficients in terms of the derivatives of at we obtain

This is the *Taylor series* for

A remarkable result: if you know the value of a well-behaved function and the values of all of its derivatives *at the single point* then you know at *all* points Besides, there is nothing special about so is also determined by its value and the values of its derivatives at any other point :