Math 128A, Spring 2016 Problem Set 06 Question 1 (a) Show that 1 xx dx = 0 nn n=1 (b) Use the sum in (a) to evaluate the integral in (a) to 12-digit accuracy. (c) Evaluate the integral in (a) by Romberg integration. Estimate how many function evaluations Romberg integration will require to achieve 12digit accuracy. Explain the agreement or disagreement of your results with theory. Question 2 In class we proved the Euler-Maclaurin summation formula 1 f (x)dx = 0 1 bm f (2m1) (1) f (2m1) (0) (f (0) + f (1)) + 2 m=1 for some unknown constants bm independent of f . (a) Find a formula for bm by evaluating both sides for f (x) = ex where is a parameter. (b) Compute b1 , b2 , b3 , . . . , b10 . Question 3 (a) Use the Euler-Maclaurin formula to show that n j k = Pk+1 (n) j=1 is a degree-(k + 1) polynomial in n. Example: n j= j=1 n(n + 1) . 2 (b) Use the results of question 2 to nd Pk+1 for 2 k 10. (c) Use polynomial interpolation to nd Pk+1 for 2 k 10 and compare with the results from (b). 1