Taylor's Theorem (in the version I'm most familiar with) states that for a function $f$ which is $n+1$ times continuously differentiable on an open neighborhood containing the segment $L$ between $a$ and $a+h$, then $$f(a+h) = f(a) + f'(a)h + \frac{1}{2!} f''(a)h^2 + \cdots + \frac{1}{n!}f^{(n)}(a)h^n + R_n(h)$$ Where $$R_n(h) = \frac{f^{(n+1)}(\xi)h^{n+1}}{(n+1)!}$$ for some $\xi \in L$. If we let $$M_n = \sup_{y \in L}\left|\frac{f^{(n+1)}(y)h^{n+1}}{(n+1)!}\right|$$ and let $f$ be infinitely differentiable in some open neighborhood of $L$, then $M_n \to 0$ implies that $$f(a+h) = \sum_{n=0}^\infty \frac{f^{(n)}(a)h^n}{n!} \tag{1}$$
I am wondering if there are any examples of functions $f$ for which $M_n \not\to 0$, but equation (1) still holds for some $a$ and $h \not= 0$.
Yes, it is possible. A simple example is $\ln 2$ from the Taylor series for $\ln (1 + x)$. In particular, $$ \ln (1 + x) = \sum_{n \geq 1} \frac{(-1)^{n + 1}x^n}{n},$$ and the remainder term does not go to $0$ when $x = 1$. But the resulting alternating series does actually converge to $\ln 2$.