# Finding the Maclaurin series of a given function knowing another Maclaurin series

Using the Maclaurin series for $\frac{1}{1-x}$, compute the Maclaurin series for: $$\frac{1+x}{1+x^2}$$.

What is the interval of convergence of this series?

Once you get the Maclaurin series for the function, I know how to find the convergence interval (I can use the Ratio test here, I believe) but how would I find the Maclaurin series for that using the one given?

$$\frac{1}{1-x}=\sum^\infty_{n=0}x^n$$

$$\dfrac{1+x}{1+x^2} = (1+x)\dfrac1{1+x^2}=(1+x)\dfrac1{1-(-x^2)}$$