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 representfunctions 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

Solution Preview

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 ...

... a zn For every power series there exist ... of the series are unbounded and the series is consequently ...Proof :- Let R be the radius of convergence which satisfies ...

... A detailed proof that the power series expansion of the given product of exponential factors is as claimed is provided, and the expansions of the individual ...

... F[[x]] be the set of all formal powers series in x ... x]] by adding the coefficients of like powers of x ... For each of the following, determine (with proof) if the ...

...Proof of Claim: ... Then in the neighborhood |z − z0 | < r2 the function f (z ) can also be represented by the power series f (z ) = Σ∞ bn (z − z0 )n n ...

... use Taylor's theorem to find a convergent power series, but this power series is not ... is zero (which is again easy) Combining these steps will proof the result. ...

... formed by the set of all formal power series p(x ... This solution provides the detailed proofs and examples of ...Proof: (⟹) Suppose that is a local ring with a ...

... the verdict you could support your opinion by highlighting the power of DNA ... There are a series of genetic syndromes where females can carry Y chromosomes (or ...

... products; established more uniform standards for proof of dumping ... name attracted investors in a series of oil ... Great power derives from the control of information ...

... What is their proof? ... is a guided step by step preparation of a possible power point presentation ... and Nutrition Board released the sixth in a series of reports ...