Modified Equation for Euler Scheme

Question

Consider the ODE $$\dot{u} = f(u)$$ with $f$ globally Lipschitz so a solution exists and is unique. I can use an explicit Euler scheme to solve this via the iterates $$u_{n+1} = u_n + hf(u_n)$$ and this numerical scheme will be consistent with my ODE to order $\mathcal{O}(h^2)$. If I instead considered the ODE $$\sum_{k=1}^p \frac{h^{k-1}}{k!} \frac{d^k u}{dt^k} = f(u) $$ then the same numerical scheme will be consistent with this ODE to order $\mathcal{O}(h^{p+1})$; this is easily verifiable via Taylor expansion. My question is what happens when we let $p \rightarrow \infty$? In particular, if I define the sequence of linear operators $T_p : C^\infty ([0,\infty); \mathbb{R}^d) \rightarrow C^\infty ([0,\infty); \mathbb{R}^d)$ via $$T_p u = \sum_{k=1}^p \frac{h^{k-1}}{k!} \frac{d^k u}{dt^k}$$ then is there a limiting operator $T$ such that $T_p \rightarrow T$ as $p \rightarrow \infty$. If there is, what can I say about it? Does it have a closed form? Intuitively, one would think that this does convernge because if we just look at the coefficients $$\sum_{k=1}^\infty \frac{h^{k-1}}{k!} = \frac{e^h - 1}{h}$$ but I'm not sure how to connect this. I can't seem to find much literature for modified equations for ODE(s), so any reference would be very much appreciated. Thank you.