I know that d/dx arctan(x) can be represented by finding its derivative, which is 1/(1+x^2). However, when I take the derivative of 2xarctan(8x), I get 2arctan(8x)+2x/(1+(8x)^2), which isn't in the correct form for a power series. So now I don't know where to go.
Thanks for helping!
yes, the reason is this. the power series in question converges to arctan(8x), and since 2xarctan(8x) = 2x*arctan(8x), we can simply replace the latter term with its power series, as it is the same thing, and then keep the 2x in front and multiply through. this works because of the convergence of the power series to the function.