(-1 )n4ntly+1 In this problem, we investigate the power series M 3n+1 (n + 1) n=0 (-1)nantlyn+1 (a) Show that 4x is the Taylor series for the function f(x) = In 3 n=0 3n+1 (n + 1) + 1 centred on x = 0. (b) For which values of x is the series equal to In 4.x 3 + 1 ) ? (-1)nan+1 xn+1 (-1 )~4n+1n+5 (c) If we multiply the series by a we get x 3n+1 (n + 1) 3n+1(n + 1) , a re- n=0 n=0 lated power series, which is equal to x* In 4x + 1 ) on the same interval you identified in part b. Express / a4 In 4x + 1 ) dx as a power series in x. (d) Let g(x) = x4 In 3 +1. Find g'(x), and the Maclaurin series for g'(x). Consider, but do not give a response to the question: should you find the Maclaurin series for g'(x) directly, or by starting from the series in (c)