This Quantum World/Appendix/Taylor series

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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 :