Let $f(x)={\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n(x-2{)}^n}{n+1}.$ Find the interval of convergence $($i$.$o$.$c$.)$ and show all steps for checking the endpoints.

a$)f(x)$

b$)$ Select ONE of the following and find the interval of convergence $($show check of endpoints$).$

i$)f^{\text{'}}(x)$

OR

ii$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(x)dx$