Consider the recurrence $f_{n+1}=f_n + \ln(f_n)$ with $f_0=2$. Also consider differential equations of type $g(0)=2$ and $\dfrac{d g}{d x}=\ln(g(x)- C \cdot \ln(g(x)))$. Lets call the solution depending on the real value $C$ as follows : $g_C(x)$. It has been shown by Did here
An issue with approximations of a recurrence sequence
that the recurrence $f(x)$ lies between $g_1(x)$ and $g_0(x)$. So my question is what is the best value for $0<C<1$ such that $g_C(x)$ is as close to f as can be for large $x$ ?