Answer
(a) Yes, it is convergent
(b) No, it does not follow that $\Sigma_{n=0}^{\infty}c_{n}(-4)^{n}$ converges.
Work Step by Step
(a) For a power series there is a positive number $R$ such that the series converges when $|x-a|\lt R$
In the given problem, $|x-a|=4$
Thus, $R\gt 4$
and the minimum interval of convergence would be $(a-4,a+4)$
Since, $|-2|=2\lt 4$ , that is within the interval of convergence for the minimum $R=4$ and it follows that $\Sigma_{n=0}^{\infty}c_{n}(-2)^{n}$ converges also.
(b) The given function could be either converge or diverge
Thus,
It does not follow that $\Sigma_{n=0}^{\infty}c_{n}(-4)^{n}$ converges.
