
power series of arctan'x
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?

If $\displaystyle x<1$ then,
$\displaystyle 1x^2+x^4x^6 + ... = \frac{1}{1+x^2}$
Integrate both side from $\displaystyle 0\mbox{ to }t$ where $\displaystyle t<1$ to get,
$\displaystyle \tan^{1}t = t  \frac{t^3}{3}+...$
Now this hold if $\displaystyle 1 < t < 1$.
But it also hold when $\displaystyle t=1$.
That is a little more difficult to show.