Problem $1.($total $10$ pts$.)$ Consider the power series $f(x)={\displaystyle \underset{n=0}{\overset{}{}}}\frac{2^n}{n^2+3}(x-1{)}^n.$

$(3$ pts$.)$ a$.$ Determine the radius of convergence.

$(3$ pt$.)$ b$.$ Determine the interval of convergence and justify your answer.

$(2$ pts$.)$ c$.$ Determine the power series for $f^{\text{'}}(x)$ and specify its radius of convergence.

$(2$ pts$.)$ d$.$ Express ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(x)dx$ as a power series.