4. (24 points) Let Si(n) = =1 k, S2(n) = %=1 k2 and Sz(n) = %=1 k*. In class, we already showed how to calculate Si(n) and S2(n). Your task here is to apply similar methods for S3(n). (a) Without calculating the closed form, use rough estimations to derive a lower bound and an upper bound for S3(n), so that you can use them to express S3(n) in the () notation. Give this e notation in the simplest form (e.g., (n) is in the simplest form but (2n) is not). (6 points) (b) Use the perturbation method (as discussed in class) to derive the closed form for S3(n). (Background Information: You would need the formula of Sin) and S2(n), which we already know: Si(n) = n(n + 1)/2 and S2(n) = n(n + 1) (2n + 1)/6.) (18 points)