How do I compute the Taylor Series for $\arctan(x)$?

Question

I've just stumbled upon Taylor Series on Wikipedia and I've been trying to obtain an expansion for $\arctan(x)$, but I can't manage to see a pattern for the $n$th derivative . Can someone come up with a solution ?

Answer 1

As suggested by David Mitra, start with $$\frac {1}{1+y}=\sum\limits_{n=0}^\infty (-1)^{n} y^n$$ Replace now $y$ by $x^2$ to get $$\frac {1}{1+x^2}=\sum\limits_{n=0}^\infty (-1)^{n} x^{2n}$$ Now, integrate both sides with respect fo $x$. The lhs is $\tan ^{-1}(x)$ and the rhs what you are looking for.