I'm trying to estimate the value of $\sqrt{5}$ using the Taylor Polynomial with the accuracy of $\frac{1}{10^3}$.

What I did was to calculate the third derivative of $\sqrt{x}$, which is $\frac{3}{8x^2\sqrt{x}}$. I calculate the third derivative, because I want accuracy on 3 digits.

Next I calculate the degree $n$ in Taylor polynomial for which I need to calculate to get the required accuracy: $|\frac{8x^2\*\sqrt{x}}{3!}|\Leftrightarrow \frac{1}{16x^2\*\sqrt{x}}\leq\frac{1}{10^3} \Rightarrow n \geq 6 $.

So I have to calculate Taylor polynomial for $\sqrt{x+1}$ of degree $6$ and plug $4$ for $x$. 

Am I doing it correctly? 

Thanks!