This Quantum World/Appendix/Taylor series

Taylor seriesEdit

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  :