1. (a) Show the equality of the formal poswer series (1-x) (1+12) = 1+x. Give a...

Transcribed Image Text:

1. (a) Show the equality of the formal poswer series (1-x) (1+12) = 1+x. Give a brief justification why this implies 1 - x2 1 1+x3 1 + 12 1-x (1) n=0 (b) Expand both sides of equation (1) as formal power series (i.e. in form of n anx"), so that comparing the coefficients of x N on both sides helps you solve part (c). Hint: for the LHS, apply the geometric series, while for the RHS, first apply the geometric series, and then the negative binomial theorem. You can leave the RHS as a double sum. (c) By comparing the coefficients of x in the answer for (b), prove that (-1)* (N k0 - k - N - 2k = if N1 mod 3 1 otherwise (2) 1. (a) Show the equality of the formal poswer series (1-x) (1+12) = 1+x. Give a brief justification why this implies 1 - x2 1 1+x3 1 + 12 1-x (1) n=0 (b) Expand both sides of equation (1) as formal power series (i.e. in form of n anx"), so that comparing the coefficients of x N on both sides helps you solve part (c). Hint: for the LHS, apply the geometric series, while for the RHS, first apply the geometric series, and then the negative binomial theorem. You can leave the RHS as a double sum. (c) By comparing the coefficients of x in the answer for (b), prove that (-1)* (N k0 - k - N - 2k = if N1 mod 3 1 otherwise (2)