1. ## Maclaurin Series: Radius of Convergence for 2x/(1+x^2)

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?

2. ## Re: Maclaurin Series: Radius of Convergence for 2x/(1+x^2)

The integral of the function is $\ln |1+x^2|$. The MacLaurin series for $\ln |1+x|$ converges for $|x| < 1$. So that of the integral converges for $x^2 < 1$.

3. ## Re: Maclaurin Series: Radius of Convergence for 2x/(1+x^2)

I once had this exact same problem on an exam! A friend did, in fact, try to find the Maclaurin series but did not have time to finish the problem. I observed that $\displaystyle 1+ x^2$ is 0 at $\displaystyle x= \pm i$, distance 1 from the origin. The radius of convergence is 1.