Let K be a field, and let D be the twisted series K[x], so that xc = σ(c)x for some field automorphism σ. Synthetic division shows D is a division ring. (You can generalize all this to twisted laurent series if you like.) Let K′ be the fixed field of σ.
First, let the order of σ be infinite. Let the series ∑ aixi lie in the center, hence it commutes with every scalar b in K. Therefore b*ai = ai*σi(b). For every i other than 0, σi moves some b in K. For this b, the equation cannot be satisfied unless ai = 0. Therefore the series is merely a constant term a0. To commute with x, a0 lies in K′. Therefore the center of D is the fixed field K′, and D is centrally infinite.
Next assume σ has order m. Commuting a series with every scalar b in K shows only powers of xm are allowed. If any coefficient lies outside of K′, the series does not commute with x. Therefore F, the center, consists of series whose exponents are multiples of m, with coefficients in K′. By galois theory, the index of K′ in K is m. Build a basis for D, as an F vector space, by crossing the basis of K over K′, with the powers of x from 0 to m-1. the dimension of D over F is m2, and D is centrally finite.