Answer Happy

1. Define R[[2]] of formal power series in the indeterminate x with coefficients from R to be all formal infinite sums Σαηα" " = 20 +ajx + a2x2 + a3x3 + oo n=0 Define addition and multiplication of power series in the same way as for power series with real or complex coefficients i.e., extend polynomial addition and multiplication to power series as though they were “polynomials of infinite degree”: IM8 anxn + 8Wi bnan " n=0 (an + bn)x" n=0 n=0 brz" Σ(Σαμόα Α) ". anxn x x n=0 n=0 n=0 k=0 (The term “formal” is used here to indicate that convergence is not considered, so that formal power series need not represent functions on R.) (a) Prove that R[[2]] is a commutative ring with 1.