1. Power Series Assignment

So my calculus professor has given us this online assignment on power series, and I can't wrap my head around in for the life of me.
Most of the questions are the same, so I think if I get the process down for one of them, then I'll be able to approach the other ones.

ex. The function f(x)= 4xarctan(2x) is represented as a power series:
(see attached image at the bottom, since I'm not sure on how to input that on the computer)

Find the first few coefficients in the power series.
c0=
c1=
c2=
c3=
c4=

Find the radius of convergence R of the series.

Any help would be greatly appreciated!!

2. You can find the series representation as follows:

$\displaystyle \arctan 2x = \int_0^{2x} {\frac{1} {{1 + u^2 }}\,du} = \int_0^{2x} {\left\{ {\sum\limits_{k = 0}^\infty {( - 1)^k u^{2k} } } \right\}\,du} .$

Now you just interchange the sum by the integral and you'll get the series for $\displaystyle \arctan2x.$ The next step is to multiply the sum (which you'll get) by $\displaystyle 4x$ and there's your series.

From there you can find its radius of convergence.