Power Series Proof
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Define the set R[[X]] of formal power series in the indeterminate X with coefficients from R to be all formal infinite sums
sum(a_nX^n)=a_0 +a_1X+a_2X^2+...
Define addition and multiplication of power series in the same way as for power series with real or complex coeficients,i.e extend polynomial addition and multiplication to power series as though they were "polynomial of infinite degree":
sum(a_nX^n)+sum(b_nX^n)=sum(a_n+b_n)X^n and
(sum(a_nX^n))(sum(b_nX^n))=sum(sum(a_kb_n-k)X^n
(P.S: The term "formal" is used here to indicate that convergence is not considered, so that formal power series need not represent functions on R)
Assuming that R[[X]] is a commutative ring with 1 prove:
a)That 1-X is a unit in R[[X]] with inverse 1+X+X^2+X^3+.....
b)That sum(a_nX^n) is a unit in R[[X]] iff a_0 is a unit in R.
The sums here go from n=0 to infinity
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Solution Summary
A power series proof is provided.
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Proof:
a) Formally, we have
(1-x)(1+x+x^2+...)
=(1+x+x^2+...) - x(1+x+x^2+...)
=(1+x+x^2+...) - (x+x^2+x^3+...)
=1+(x+x^2+...) - (x+x^2+...) = 1
Thus (1-x) is a unit and its inverse is 1+x+x^2+...
b) "=>": If ...
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