I'm trying to figure out if there is a better way to teach the following Taylor series problem. I can do the problem myself, but my solution doesn't seem very nice!
Let's say I want to find the first $n$ terms (small $n$ – say 3 or 4) in the Taylor series for
$$
f(z) = \frac{1}{1+z^2}
$$
around $z_0 = 2$ (or more generally around any $z_0\neq 0$, to make it interesting!) Obviously, two methods that come to mind are 1) computing the derivatives $f^{(n)}(z_0)$, which quickly turns into a bit of a mess, and 2) making a change of variables $w = z-z_0$, then computing the power series expansion for
$$
g(w) = \frac{1}{1+(w+z_0)^2}
$$ and trying to simplify it, which also turns into a bit of a mess. Neither approach seems particularly rapid or elegant. Any thoughts?
Best Answer
Let $g(w) = \sum_{n=0}^{\infty} a_n w^n$.
Then $(w^2+4w+5) \; g(w) = 1$ implies $$\begin{align} 5 a_0 &= 1 \\ 4 a_0 + 5 a_1 &= 0 \\ a_0 + 4 a_1 + 5 a_2 &= 0 \\ a_1 + 4 a_2 + 5 a_3 &= 0 \\ \text{etc.} \end{align}$$
which you can then solve for the $a_n$'s in a stepwise fashion.