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.

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 ...

...series method and are therefore typically expressed as power series. ... case when ν = ±1/2, the series solutions can ... 3) in conjunction with a proof by induction ...

...Proof: First, I claim that the field of fractions of F[[x]] is F[[x]] itself. Since F[[x]] is the ring of formal power series, then without considering ...

... k = 0 k! doesn't contain negative powers of (s ... 1)! s = s0 1! 2! and our proof is completely ... to determine a residue by means of Laurent series, we need ...

...series and EMBED Equation.3 is the power series obtained by ... has the same radius of convergence as the original series. Proof :- Let R and R 2 be the radii of ...

... The solution provides proof in number in numerical analysis ... If we truncate the series after the quadratic term it ... is some polynomial with the lowest power of h ...

... This shows how to use the ratio test to determine when the power series converges, and complete a proof regarding the Taylor series that involves the Fibonacci ...

... 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 ...