I was having trouble with this problem:
Use a power series to approximate the value of the integral with an error less than 0.0001. Assume that the integrand is defined as 1 when x=0.
I think using geometric series would be the way to go because finding the derivatives of arctan(x) and generalizing looks like it's going to be a pain.
Recall the geometric series:
Looking at the derivative of arctan(x), we have which resembles the geometric series with
This gives us:
Integrating gives you the power series for arctan(x), then the rest follows as TheEmptySet showed you.
If you need to find the maclaurin series for arctan x by finding all the derivatives, it's done here: Maclaruin Series (Scroll down a bit)
Gets messy ...