When a Function Does Not Equal Its Taylor Series
Not every function is analytic. The video below explores
the different ways in which a Taylor series can fail to converge
to a function $f(x)$.
 The function may not be infinitely differentiable,
so the Taylor series may not even be defined.
 The derivatives of $f(x)$ at $x=a$ may grow so
quickly that the Taylor series may not converge.
 The series may converge to something other than
$f(x)$.

