I'm stumped here. I''m supposed to find the Maclaurin series of $\frac1{1+x^2}$, but I'm not sure what to do. I know the general idea: find $\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$. What I want to do is find some derivatives and try to see if there's a pattern to their value at $0$. But after the second derivative or so, it becomes pretty difficult to continue. I know this:
$$f(0) = 1$$
$$f'(0) = 0$$
$$f''(0) = -2$$
$$f^{(3)}(0) = 0$$
$$f^{(4)}(0) = 0$$
But when trying to calculate the fifth derivative, I sort of gave up, because it was becoming too unwieldly, and I didn't even know if I was going somewhere with this, not to mention the high probability of making a mistake while differentiating.
Is there a better of way of doing this? Differentiaing many times and then trying to find a pattern doesn't seem to be working.