how could i expand something such as arctan'x (derivate of arctanx ... i.e. d/dx arctanx) into a power series. also how would you be able to find the radius of convergence for it?

so far i have managed to work out that:

arctan'x = $\displaystyle \frac{1}{1 + x^2} $

$\displaystyle \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 +...+ (- 1)^n x^{2n}$

how do you work out the "radius of convergence" though: i know it is : |x|< 1.. but how do you work it out please?

i tried it on $\displaystyle (-1)^n x^{2n}$

i ended up with

$\displaystyle a_{n+1} / a_{n} = \frac{|x|^{2n + 2}}{|x|^{2n}} = |x|^2/1 $ as n tends to infinity... ...

so radius of convergence is |x|< 1...

is this working out correct?