Part of my assignment is to find the third degree Taylor Series of $\tan(x)$ about $\pi/4$ and then estimate the error of this approximation when evaluated at 0.75.
Finding the series was easy enough, but I wasn't sure about the error estimate.
From what I know the error is:
$$E_n \leq \frac{| f ^{(n+1)} (c) |\cdot|x-a|^{(n+1)}}{ (n+1)! } ,$$
where $c$ is between $x$ and $a$.
Now, when I compute this for $\tan{(x)}$ I end up with:
$$ E_n \leq \frac{|-4[\cos(2c)-5]\tan(c)\sec^4(c)|\cdot|0.75 - \pi/4|^4} {24} $$
Not only is the fourth derivative of tan(x) really messy, I have no idea what $c$ to choose so that I can simplify things nicely. The whole point was to estimate the error for tan but tan shows up in the error estimate itself! Does this $c$ have to be strictly between the center of the series and the point I'm evaluating at?