I'm trying to figure out the rate of the remainder term in the Taylor expansion of some functional $F$ using its Gateaux derivative. I found the following from Wikipedia, where $U$ is an open subset of some locally convex topological vector space.
Suppose that the line segment between $u \in U$ and $u + h$ lies entirely within $U$. If $F$ is $C^k$ (in the sense that its $k$-th Gateaux derivative is a continuous function) then $$F(u+h)=F(u)+dF(u;h)+\frac{1}{2!}d^2F(u;h)+\dots+\frac{1}{(k-1)!}d^{k-1}F(u;h)+R_k$$ where the remainder term is given by $$R_k(u;h)=\frac{1}{(k-1)!}\int_0^1(1-t)^{k-1}d^kF(u+th;h)\,dt.$$
I'm wondering under what conditions can we claim that the remainder term $R_k$ in this case is $O(h^k)$, as in the Taylor's theorem for a univariate function, or some other $O(\cdot)$ that's related to $h$? We know that the $k$-th Gateaux derivative is homogeneous of degree $k$ in $h$, so there should be a factor of $h^k$ in the rate. But I'm not sure how the fact that we're integrating over $t$ affects the derivation of the rate.
I searched online trying to find some references but without much luck...Any help would be greatly appreciated!