Power series representation

Find the power series representation for each of the following functions. Include the interval of convergence.

 1. \(\displaystyle{f(x)= \frac{2}{1+x}}\)   solution

 2. \(\displaystyle{f(x)= \frac{3}{2-x}}\)  solution

 3. \(\displaystyle{f(x)= \frac{x}{1-2x}}\)   solution

 4. \(\displaystyle{ f(x)=\frac{5x}{x^2+4}}\)  solution

 5. \(\displaystyle{f(x)=\frac{1+x}{1-x}}\)  solution

 6. \(\displaystyle{ f(x)=\frac{1}{(1-x)^2}}\)  solution

 7. \(\displaystyle{ f(x)= \frac{x}{(1+x)^2}} \)  solution

 8. \(\displaystyle{ f(x)= x^2 \tan^{-1} x} \)  solution

 9. \(\displaystyle{ f(x)= \frac{\tan^{-1} (2x)}{x^2}} \)   solution

 10. \(\displaystyle{ f(x)=\ln(5-x)}\)   solution

 11. \(\displaystyle{f(x)=\ln(2+x)}\) solution

 12. \(\displaystyle{ f(x)=\ln|2+3x|}\)  solution    Alt Solution