I have to write an essay on the flat function
$$\text{flat}(x) = \begin{cases} e^{-\frac{1}{x^2}} & \text{for } x \ne 0 \\ 0 & \text{for } x = 0 \end{cases}$$
and I want to prove that the Maclauren series of flat does not converge to flat.
Can anyone help me with this?
http://homepages.math.uic.edu/~fthulin/essay3math.pdf
This is what my professor wrote about it... But I don't understand it...
It is clearly written in the file you linked that the "Proposition" follows from the definition of the Maclaurin series (and Lemma 2).
It is shown in Lemma 2 that $f^{(n)}(0)=0$ for all $n$. (For simplicity I denote $f(x)=\textrm{flat}(x)$.) By the definition of the Maclaurin series of $f$ at $x=0$, the series is zero because all the its coefficits are zero. But $f$ is not zero for all $x\neq 0$. So they cannot be equal.