Define the set R[[x]] of formal power series in the indeterminate x with coefficients from
R to be all formal infinite sums
00
Lanxn = ao + a1x + a2x
2 + a3
x3 + · · ·.
n=O
Define addition and multiplication of power series in the same way as for power series
with real or complex coefficients i.e., extend polynomial addition and multiplication to
power series as though they were "polynomials of infinite degree":
00
Lanx
n
n=O
00
LanX
n
n=O
+
X
00 00
Lb
n
x
n = L(an + bn)Xn
oo oo n
L bnx
n = L (Lakbn-k)x
n
.
n=O n=O k=O
(The term "formal" is used here to indicate that convergence is not considered, so that
formal power series need not represent functions on R.)
(a) Prove that R[[x]] is a commutative ring with 1.