Question: Consider the function $\displaystyle \frac{2x}{1+x^2}$. What is the radius of convergence for the Maclaurin Series approximating this function?

My thoughts: I can take the derivative a bunch of times to recognize a pattern. I find that the series essentially becomes an alternating geometric series. I can then describe the series using summation notation and analyze this to get a radius of convergence of 1. But I don't think this is the best method to solve the problem since it involves so much time (quotient rule gets progressively more cumbersome).

Does anyone have a better method to easily identify the series?