Infinite Sequences and Series · Hard problem

A polynomial approximation $P_n$ for $f(x)=\ln(1+x)$, built from derivatives at $0$, is used to estimate $\ln(1.5)$. Given that $\left|f^{(n+1)}(c)\right|\le n!$ for every $c$ between $0$ and $0.5$, the resulting error is guaranteed to be at most $\dfrac{(0.5)^{n+1}}{n+1}$. Find the smallest $n$ for which this guarantee is below $10^{-4}$.