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


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