I am using the formula: $f(x)=\frac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}$
Is it correct to use this formula for our function like this:
$f(x)=\frac{1}{4+x^{2}}=\frac{1}{1-(-x^{2}-3)}=\sum_{n=0}^{\infty}(-x^{2}-3)^{n}$
Also, how can we determine radius of convergence without any specific calculations?
I know that in order for the series to be convergent $|-x^{2}-3|<1$ should be true, does this mean that radius $R=1$?
$$\frac{1}{4+x^2} = \frac{1}{4}\left(\frac{1}{1+\frac{x^2}{4}}\right) = \frac{1}{4}\left( \frac{1}{1 - \left(-\left(\frac{x}{2}\right)^2\right)} \right).$$ Therefore, if $$\frac{1}{1-z} = \sum_{k=0}^\infty z^k,$$ the choice $z = -(x/2)^2$ yields the desired series expansion about $x = 0$.