See attached.

Problem 6: 15 points. Consider f(x) = xe* + 1. (a) Give a formula for the nth Taylor remainder Rn (x) of f, centered at x = 0. (b) Show limn +0. Rn(x) = 0 for all r ER. Problem 7: 15 points. For any a E R, define the binomial-type coefficient by C(a, n) _ a(a - 1)(a - 2) ...(a - n + 1) n! for n > 1 and C(a, 0) = 1. For example, C(1, 2) = =1 Let g(x) = >x (a) Determine the radius of convergence r for the power series g(x). (b) Determine the power series for the derivative # g(x) on the interval (-r, r). (c) Show &g(x) = } En C(), n)x" by examining the coefficients in part (b)