Many functions have power series expansions.
In order to verify that these function are equal to power series we will need to know in advance what power series we are trying to show the function is equal to. If a function can be expressed as a power series then the values of that function are completely determined throughout its interval of convergence by the values of its derivatives at a single point . In other words, the behavior of the function at just one point is enough to determine how it will continue to behave, even at points which are far away. Clearly functions of this kind are particularly well-behaved. Although it is beyond the scope of this course, it is possible to identify which functions are expressible as power series, and the answer comes from complex analysis.