Does a Taylor series always converge to its generating function? Can you please explain?
Also, I've encountered an exercise in my Math book.
What is the Taylor series generated by a function $ f = f(x)$ at a point $x = a$?
What information do you need about f to construct the series? Give an example.
Thank you very much in advance! I'll be forever thankful!
Best Answer
In general: no! A real valued function is called analytic if it is given by a convergent power series. (A Taylor series being an example of a power series.) The canonical counter-example is $f(x) = e^{-1/x^2}.$ You find that $f(x) \to 0$ as $x \to 0$ and, moreover, all of its derivatives tend to zero as $x \to 0$. The Taylor series of $f(x) = e^{-1/x^2}$ at $x=0$ is identically zero, while the function is not. Such a function is called a flat function. In fact, analytic functions form a very "small" subset of functions.