Evaluate the integral in a by romberg integration estimate

Question

Question 1 (a) Show that

₀∫¹x^-x dx = _n=1Σ^∞ n^-n

(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 12-digit accuracy. Explain the agreement or disagreement of your results with theory.

Question 2 In class we proved the Euler-Maclaurin summation formula

₀∫¹ f(x) dx = ½ (f(0) + f(1)) + _m=1Σ^∞ b_m (f^(2m-1) (1) - f^(2m-1) (0))

for some unknown constants b_m independent of f.

(a) Find a formula for b_m by evaluating both sides for f(x) = e^λx where λ is a parameter.

(b) Compute b_1, b_2, b_3,....., b₁₀.

Question 3 (a) Use the Euler-Maclaurin formula to show that

_j=1Σⁿ j^k = P_k+1(n)

is a degree-(k + 1) polynomial in n. Example:

_j=1Σⁿ j= n(n + 1) / 2.

b) Use the results of question 2 to find P_k+1 for 2 ≤ k ≤ 10.

(c) Use polynomial interpolation to find P_k+1 for 2 ≤ k ≤ 10 and compare with the results from (b).