Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence. Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence. My problem here is the Taylor series. Computing the few first derivative is possible, but I can't seem to have caught any pattern I can formulate elegantly. Maybe I should only make an observation about the derivatives without using a formula? I could really use any help here.
Edit: I arrived at: $$f(x)=-2\sum_{n=0}^{\infty}{x^{2+4n}\over 2n+1}$$. Using the ratio test: $$\lim\limits_{n\to \infty}|{a_{n+1}\over a_n}|=\lim\limits_{n\to \infty}|{{x^2x^{4n}x^4\over 2n+3}\over {x^2x^{4n}\over 2n+1}}|=\lim\limits_{n\to \infty}|{x^4(2n+1)\over 2n+3}|=|x^4|$$. $|x^4|<1\iff |x|<1 \Rightarrow R=1$.

Outline: Write down the Maclaurin expansion of $\ln(1-x^2)$, of $\ln(1+x^2)$, and subtract term by term.  For $\ln(1+x^2)$, write down the series for $\ln(1+t)$ and replace $t$ by $x^2$.
The radius of convergence can be found in one of the usual ways, for example by using the Ratio Test.
