Spring 1998

Maple Lab, Week 29

Far-Out Integrals II

Based on suggestions by M. L. Platt, CASE Newsletter #28, April 1997

References: Handout for Lab 21 (Far-Out Integrals I); Improper integrals: Stewart Sec. 7.9; Infinite series: Stewart Chap. 10

Theme: We continue our investigation of I ≡ ∫_{0}^{∞} \frac{sin x}{x} dx.

Exercises: 5. How do we know that the improper integral I is convergent? Study the Comparison Theorem for Integrals, Stewart pp. 493–494. Although Stewart never gets around to saying so, the series theorem on p. 635 has an analogue for improper integrals:

If ∫_{a}^{∞} |f(x)| dx converges, then ∫_{a}^{∞} f(x) dx converges.

Therefore, the condition f(x) ≥ g(x) ≥ 0 in the comparison theorem can be replaced by f(x) ≥ |g(x)|.

(A) Can you apply the comparison theorem directly to the formula above for I? (Why not?)

(B) Integrate by parts to get an integral to which the comparison theorem applies. Hint:

∫_{0}^{∞} f(x) dx = ∫_{0}^{π} f(x) dx + ∫_{π}^{∞} f(x) dx.

6. Define a_{k} = ∫_{kπ}^{(k+1)π} \frac{sin x}{x} dx, and consider S = ∑_{k=0}^{∞} a_{k}.

(A) Prove that the series S converges, and that the sum equals I.

(B) Write a Maple procedure to evaluate a_{k} numerically for any given k ≥ 1. (The procedure may call simpson from the student package.)

(C) Evaluate a_{0} separately. (You probably already did this if you finished the extra credit Exercise 4 in Part I.)

(D) Add up enough terms in the series S to approximate I to 2 decimal places. Hint: Alternating Series Estimation Theorem, p. 632.

7 (extra credit). Get a better approximation to I with less computation, using the sequence

b_{k} = ∫_{kπ}^{(k+1)π} \frac{cos x}{x^{2}} dx.

8 (extra credit). Is the series S absolutely convergent? After deciding, make your answer to 5(A) more precise.