My question:
Characterize those power series $\sum_{k=0}^\infty a_{k}(x-c)^{k}$ that converge uniformly on ($-\infty, \,\infty$). What does it mean to characterize a power series?
Let {$a_{k}$} be a sequence of coefficients for a power series. By definition, the radius of convergence $R = \infty$ if $\limsup\sqrt[k]{|a_{k}|} = 0$.
These power series are precisely the polynomials.
The partial sums of a uniformly convergent power series must be a Cauchy sequence in the sup-norm. Let $S_n(x)$ be the $n$-th partial sum. Uniform convergence implies
$$
\forall \epsilon > 0 \, \exists N > 0 \, \ni \, \forall n > N \, \forall x \in \mathbb{R} \,
|S_{n+1}(x) - S_n(x)| = |a_{n+1}(x-c)^{n+1}| < \epsilon
$$
Now this is only possible if $a_{n+1} = 0$. Therefore the condition becomes
$$
\exists N > 0 \, \ni \, \forall n > N \, a_{n+1} = 0
$$
i.e. this is a polynomial.
A characterization in this case is a condition on the series (that is, on its coefficients) that is equivalent to uniform convergence in the whole real line.
