# Calculate the necessary degree of expansion for a Taylor series

I've had a couple of tasks that say something along the line of "calculate the degree of the Taylor polynomial so that the difference between it and the function is such and such". I don't know any way of getting to that point without trial and error, my intuitive guesses are usually off and it takes a lot of time to calculate the difference properly since we aren't allowed to use calculators. Here's an example:

Let $f:[0,\infty)\to\Bbb{R}$ be defined by $f(x)=(x-1)^2e^{-2x}$. Calculate $n\in\Bbb{N}$ such that $T_{n,2}$, the Taylor polynomial of $n$-th order around the expansion point $x=2$, fulfills $$|f(x)-T_{n,2}(x)|\le10^{-3}$$ for all $x$ with $|x-2|\lt10^{-1}$.

I genuinely don't know how to approach this without just trying out different values for $n$, and I was wondering if there are any systematic ways to get a result. Any help would be appreciated.

• You can use Taylor-Lagrange inequality for the remainder: if you can bound the derivatives at all orders, it enables you to find an estimate for this degree. – Bernard Sep 3 '18 at 11:50
• I didn't know you had to consider the remainder for this, thank you! – pavus Sep 3 '18 at 12:02