Difference between Taylor Series and Formal Taylor Series

Question

I´ve read a bit about the difference, but i´m not sure if i´m getting it right. I understood that when you use the formal series, you can evaluate the series in some $X$ that is not necesarrly a number. The book used and example with $f(x) = e^{-1/x^2}$. Acording to the textbook, despite the fact that $f(x)$ is not continious on $x=0$, you could do the formal series exapansion of $f(X)$ where $ X = -1/x^2$.

Answer 1

The term formal Taylor series is used when you want to insist that you do not address convergence questions. The Taylor series of a function at a point is a formal power series defined for any infinitely differentiable function. However, a function may not be equal to the infinite sum defined by its Taylor series, even if its Taylor series converges at every point.