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\line{\h  Spring 1998}
\line{\h \bigbf Maple Lab, Week 29 \h}
\vss
\line{\h \bigbf Far-Out Integrals II\h}
 \vs
 \centerline{\it Based on suggestions by M. L. Platt, CASE 
Newsletter \#28, April 1997}
\vss
 {\bf References:} 
\hbox{\vtop{\hsize=5truein\baselineskip=13pt
\obeylines% ALL STEWART REFS ARE TO ED 3!
Handout for Lab 21 (Far-Out Integrals I);
Improper integrals: Stewart Sec.~7.9; 
Infinite series: Stewart Chap.~10}}
 \vss

{\bf Theme:} We continue our investigation of 
$I \equiv \d \int_0^\infty {\sin x\over x}\, dx$. 
 \vss

{\bf 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:
$$\hbox{\rm If $\d\int_a^\infty |f(x)|\, dx$ converges, then
$\d\int_a^\infty f(x)\, dx$ converges.}$$
 Therefore, the condition $f(x) \ge g(x) \ge 0$ 
 in the comparison theorem can be replaced by $f(x) \ge |g(x)|$.

 \vs
{\bf (A)} Can you apply the comparison theorem directly to the 
formula above for~$I$?   (Why not?)
 \vs

 {\bf (B)} Integrate by parts to get an integral to which the 
comparison theorem applies.
 {\sl Hint:\/} 
 $$\int_0^\infty f(x)\,dx = 
 \int_0^\pi f(x)\,dx  + \int_\pi^\infty f(x)\,dx.$$


\vss
{\bf 6.}  Define 
$\d a_k = \int_{k\pi}^{(k+1)\pi} {\sin x\over x}\, dx$,
 and consider $\d S = \sum_{k=0}^\infty a_k\,$.

 \vs
 {\bf (A)} Prove that the series $S$ converges, and that the sum 
equals~$I$.

 \vs
 {\bf (B)} Write a Maple procedure to evaluate $a_k$ numerically 
for any given~$k\ge 1$. (The procedure may call {\tt simpson} from the 
 {\tt student} package.)

 \vs
 {\bf (C)} Evaluate $a_0$ separately.  (You probably already did 
this if you finished the extra credit Exercise 4 in Part I.)
 \vs

 {\bf (D)} Add up enough terms in the series $S$ to approximate $I$ 
to 2 decimal places.  {\sl Hint:\/} Alternating Series Estimation 
Theorem, p.~632.
 
 \vss \goodbreak


 {\bf 7 (extra credit).}  Get a better approximation to $I$ 
with less computation,
using the sequence
 $$b_k = \int_{k\pi}^{(k+1)\pi} {\cos x\over x^2}\, dx.$$
\vss

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

\bye