Are there any $f(x)$ whose $\xi_n$ of the Lagrange's remainder does not converge to $0$?

Consider the Maclaurin's series of an analytic function $f$ with Lagrange's remainder.

Define $\xi_n$ as$$f(1)=\sum_{k=0}^{n-1}\frac{f^{(k)}(0)}{k!}+\frac{f^{(n)}(\xi_n)}{n!}$$where $0<\xi_n<1$, and if there are several that satisfy this, choose the one closest to $0$ (or the infimum if there are infinitely many).

Is there an example of $f$ such that $\xi_n$ doesn't converge to $0$?

I'm not sure why, but all the ones I tried seemed to go to zero.

Edit:$\frac{f^{(n)}(\xi_n)}{f^{(n)}(0)}=1+\frac{1}{n+1}\frac{f^{(n+1)}(\xi_{n+1})}{f^{(n)}(0)}$, so, if $\frac{f^{(n+1)}(\xi_{n+1})}{f^{(n)}(0)}=o(n)$ then $f^{(n)}(\xi_n)\to f^{(n)}(0)$. This implies that $\xi_n$ converges to $0$ for some functions such that $e^{ax}, \sin ax$. From the results of numerical methods for other functions, I suppose that the same is true for all analytic functions.

Update:Here are some numerical results. Some results are not shown for all $n\leq100$ because it takes too long to compute. We can see that $\xi_n$ seems to be gradually decreasing for all the functions I tried here. I hope this helps.