Problem $3$: $[$up to $10pts]$ The Taylor series for a certain function $f(x)$ centered at $x=c$ is ${\displaystyle \underset{k=0}{\overset{}{}}}\frac{2(k+1{)}^3}{4^k}(x-1{)}^{2k}.$

A$.[$up to $3$ pts$]$ Answer the following questions.

$-$ State the center of the series. $c=$

$-$ Calculate $f(1).$

B$.[$up to $4pts]$ Find the open interval of convergence for the power series. Make sure to follow the guidelines on Page $3$ of the coversheet if evaluating any limits is part of your solution.

C$.[$up to $3$ pts$]$ Explain whether the ${40}^{th}$ degree Taylor polynomial should provide better or worse approximation for $f(2)$ than the fourth degree polynomial. What about for $f(-2)?$

Can someone please help me with part A$,$ B$,$ and C on this problem?