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> How to find the Taylor series expansion of $$f(x)=\frac{1}{1+x^2}$$ about a point $a \in \Bbb{R}$ ?
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I know it has the form $$\sum\_{n=0}^\infty a\_n(x-a)^n$$ where $a\_n=\frac{f^{(n)}(a)}{n!}$.

So in order to find that expression, we find all derivatives of $f$ and then use $a\_n=\frac{f^{(n)}(a)}{n!}$ 

But finding all derivative of $f$ is a difficult task. 

Is there any other way to show this? 

What I'm misunderstanding the given $a\_n=\frac{f^{(n)}(a)}{n!}$ method (if any) ?