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Question 5 Consider the series of functions x 1+ 3 (11) n In +1 n=1 Determine the largest possible domain D for which the series converges pointwise for XED and determine whether it is true that 00 x xn+1 xn+ d dx G 1) - 2 G 111) dxn dxn n n+ +1 n=1 n=1 for all x ED. (8 marks) (b) Define a sequence of functions {In(x)}, where f(x) 2x + nx3 1 + nx2 Show that fn(x)} converges uniformly on R to a continuous function f(x), and determine if it is true that f'(x) = lim lim lat (fr) 00 Ldx for all x ER (9 marks) (c) Let {9n(x)} be a sequence of functions, where each function In (x) is monotone increasing on the interval [0,1], and {9.(x)} converges pointwise to a constant function g(x) = K on [0,1]. Show that the convergence of {9n(x)} is uniform on (0,1). (5 marks)