Master Dissertation

Theorem 1.6. Let anX n be a formal power series. Then anX n has
a unique inverse if and only if a0 has a multiplicative inverse in R.
Proof. Say, bnX n is the inverse of anX n . Then by the definition of
multiplication we must have
(i) a0b0 = 1
(ii) a0b1 + a1b0 = 0
(iii) a0b2 + a1b1 + a2b0 = 0.
and so on. . .
Now it is clear from (i) that if the inverse exists then a0 must also be
invertible.
If on the other hand a0 ∈ R ∗ , the invertible elements of R, then the inverse
of anX n is given by bnX n , where b0 = a −1
0 and
bn = a −1
0 (−a1bn−1 − . . . − anb0), for n ≥ 1.
Say, both bnX n and cnX n are inverse to anX n . Then a0b0 = a0c0,
that is b0 = c0, since a0 = 0. Assume bk = ck, for all k < n, then
a0bn + a1bn−1 + · · · + anb0 = a0cn + a1cn−1 + · · · + anc0
By the induction assumption bk = ck for all k < n hence a0bn = a0cn and
thus bn = cn.
The geometric series formula is valid in R[[X]].
Theorem 1.7. Let anX n = 1 be a formal power series, then
anX n m
= (1 − anX n ) −1 .
Proof. Let g =
anX n
m . Then
1 − anx n
g = 1 − anx n
as wanted.
lim
M→∞
m=0
M
anx n ) m
= lim 1 −
M→∞
anx n M
m=0
= 1 − anx n
+ anx n −
= 1.
anx n ) m
Setting an = δ1,n we get the usual geometric series formula.
10
anx n 2
+ · · ·