1. ## taylor

when you say that a function which is m times continuously differentiable is analytic, does it mean that in the taylor formula, the remainder tends to 0 as m tends to infinity?

2. A function is analytic if it has a convergent taylor series in a particular domain.

3. Originally Posted by alexandrabel90
when you say that a function which is m times continuously differentiable is analytic, does it mean that in the taylor formula, the remainder tends to 0 as m tends to infinity?
If the function is $C^{\infty}$ this is what it means.

4. Originally Posted by Prove It
A function is analytic if it has a convergent taylor series in a particular domain.
A convergent Taylor series which actually converges to the function! Take for example the classic $\displaystyle f(x)=\begin{cases}e^{\frac{-1}{x^2}} & \mbox{if}\quad x>0\\ 0 & \mbox{if}\quad x\leqslant 0\end{cases}$. Then, $f$ is infinitely differentiable and its Maclaurin series congerges, but it doesn't converge to $f$!

5. Originally Posted by alexandrabel90
when you say that a function which is m times continuously differentiable is analytic, does it mean that in the taylor formula, the remainder tends to 0 as m tends to infinity?
An important detail is that if a complex function is differentiable in $z=z_{0}$, then it is analytic in $z=z_{0}$ and it is representable as Taylor function 'somewhere around' $z=z_{0}$... that isn't necessarly true for an 'ordinary' real function...

Kind regards

$\chi$ $\sigma$