Consider the function f(x)=1/(1+x^2) on the real line. Using the geometric progression formula, you can expand f(x)=1-x^2+... . This series converges for |x|<1 but diverges for all other x. Why this is so? The function looks nice and smooth everywhere on the real line.
This example is taken from the Introduction of the textbook by B. V. Shabat.