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 =

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

i ended up with

as n tends to infinity... ...

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

is this working out correct?