Answer
$n=3$
Work Step by Step
The error $R_n$ can be computed as:
$|R_n(n)|=|\dfrac{f^{n+1}(z)(x-c)^{n+1}}{(n+1)!}|$
Here, we have: $|R_n(0.25)|=\dfrac{(0.25)^{n+1}}{(1+z)^{n+1} (n+1)}$
or, $\dfrac{(0.25)^{n+1}}{(n+1)} \lt 0.001$
Now, we will use the trail and error method by plugging the different values of $n$.
Thus, $|R_2(2)|=\dfrac{(0.25)^{2+1}}{(2+1)} \approx 0.005$
$|R_3(3)|=\dfrac{(0.25)^{3+1}}{(3+1)} \approx 0.00098$
We can see that the error is less than $0.001$ for the value of $n=3$.
