Answer by Auslander for Finding Taylor polynomial for fractional function

Here are a few pointers:

1. First, find the Taylor polynomial for $\frac{1}{1-x}$. The geometric series should be of some use! And you can find an explicit formula for the error term too.
2. Use this to find the Taylor polynomial for $\frac{1}{1+x^2}$. Just replace the symbol $x$ with the symbol $-x^2$.
3. Use this to find the Taylor polynomial for $\frac{1}{1+(x/3)^2}$. Again, just replace the symbol $x$ with the symbol $x/3$.
4. Now just multiply by $x/9$.

Now just determine for which values of $x$ your error term goes to zero as $n\to\infty$.