There is a proof of the theorem for $R[[x]]$ that uses the well-known result of I.S. Cohen that a ring is noetherian if and only if its prime ideals are finitely generated. Such a proof is given by Kaplansky in his 1970 book Commutative Rings, Theorem 70.

This method of proof can be generalized to noncommutative rings to show that the power series ring $R[[x]]$ over a left noetherian ring is left noetherian, as long as one is careful when putting forth a definition of "prime left ideal." Gerhard Michler gave such a proof in a paper that's not so easy to locate (but it seems to be described here), and I gave a similar proof in Theorem 3.9 in this paper.