Here is my question:
What degree Maclaurin polynomial of $e^x$ must be taken to guarantee an estimate of $e$ to within $1 \times 10^{-6}?$
I know that the error term is: $$R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1} \qquad \text{for some } c \in]0,x[$$
Here is what I did:
Since we know that $\frac{d}{dx}e^x=e^x$, we have:
$$R_n(x)=\frac{e^c}{(n+1)!}x^{n+1} < 10^{-6}$$
Now, since $c \in]0,x[$, I think $c$ should be as big as possible, but I am not sure what value to pick for $x$.