Maclaurin series of $\frac{1}{1+x^2}$ 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.

$1$) Write down the series for $\frac{1}{1-t}$. You have probably have already seen this one. If not, it can be computed by the method you were using on $\frac{1}{1+x^2}$. The derivatives are a lot easier to get a handle on than the derivatives of $\frac{1}{1+x^2}$.

$2$) Substitute $-x^2$ for $t$, and simplify.

**Comment:** It can be quite difficult to find an expression for the $n$-th derivative of a function. In many cases, we obtain the power series for a function by "recycling" known results. In particular, we often get new series by adding known ones, or by differentiating or integrating known ones term by term. Occasionally, substitution is useful.