00 . . . . (_1)n4n+1$n+1 In thls problem, we Investigate the power series ; 3n+1(n + 1) . 0 (_1)n4n+1xn+1 43: (a) Show that E m is the Taylor series for the function at) = 1n (3 + 1) n=0 centred on :r = 0. b For which values of a: is the series e ual to In E + 1 ? ( ) q 3 _1 n4n+1 n+1 _1 n4n+1 n+5 ( ) 33 Z i ) 33 i a re_ 11-0 00 . . 4 4 = (c) If we multiply the series by m we get m \":0 3M1 (n + 1) _ 3M1 (n + 1) 4:2: lated power series, which is equal to 1:4 1n (? + l) on the same interval you identied in part b. 4 Express / 3:41n (g + 1) dsc as a power series in 3:. (d) Let g(a:) = 3:4 1n (4% + 1). Find g'(a:), and the Maclaurin series for g'(m). Consider, but do not give a response to the question: should you nd the Maclaurin series for g'(m) directly, or by starting from the series in (c)