Not so much a question. Rather, I don't quite understand the concept. R[[X]] is the formal power series a0 + a1x + a2x^2.... R[X] consists of all elements in R[[X]] which have only finitely many non-zero coefficients. Can someone give me an example? Would, say, two different elements of R[[X]] be 1+x and 3+2x+x^3 and hence both of these polynomials would be in R[X]? When does R[[X]] ever have infinitely many non-zero coefficients?