help with radius of convergence! how do i do the following calc problem?

Question

The function y(x) is given by

y(x)=summation(n=0 to infinity) [x^(2n)]/(1-2n)

a) find the radius of convergence for this power series

b) show that y satisfies the differential equation
(1-x^2) y''(x) - 2xy'(x) + 2y(x)=0

Answer 1

y(x) = 1- x ArcTanh(x), so that the "radius of convergence" is -1 < x < 1, and plugging in y(x) into the differential equation (1-x^2) y''(x) - 2xy'(x) + 2y(x) gets you 0.

You can also plug in the series into the differential equation, and come up with 2(n+1) x^(2n), which is the same as saying the series converges as n -> ∞, and that the equation = 0. However, solving the differential equation to get 1 - x ArcTanh(x) is not easy, even though it's another way of finding the sum of the series.