III. 2.Provethat if Summation from k = 1 to infinity of g k converges uniformly over the intervalI, then Summation from k = 1 to infinity of f k (x) = (x-a) summation from k = 1 to infinityg k(x) also converges uniformly overI.

III.1. Give an example of a function series for which each summand fx is differentiable at everyxin an intervalIand Summation from k = 1 to infinity of fk' (the sum of the derivatives of the original summands) converges uniformly overI, but Summation from k = 1 to infinity of fk(x) does not converge pointwise for anyxinI.Prove that your function series satisfies all these conditions.