Talk:Taylor series

"Note that there are examples of infinitely often differentiable functions f(x) whose Taylor series converge but are not equal to f(x). For instance, all the derivatives of f(x) = exp(-1/x²) are zero at x = 0, so the Taylor series of f(x) is zero, and its radius of convergence is infinite, even though the function most definitely is not zero."

f(x) has no Taylor series for a=0, since f(0) is not defined. You have to state explicitly that you've defined f(x)=exp(-1/x²) for x not equal to 0 and f(0)=0 . This is merely lim[x->0] f(x), but it is a requirement for rigor.

Don't complain, fix! Wikipedia:Be bold in editing pages. -- Tim Starling 02:03 16 Jun 2003 (UTC)