COALITION MATH 152, SPRING 1998

                         (S. A. Fulling)

                             Day 28.T


    1.  The usual paper shuffling:  Old homework, CAPA, lab 
    sheet, handout on testing series for convergence.  Announce 
    test stats.  Settle whether last Tuesday morning will be used 
    as a review session (before the English class dealing with 
    the Grabiner article).
SINCE THE GRADER WAS NOT PRESENT, HOMEWORK AND CAPA-TROUBLE 
COLLECTION HAD TO BE POSTPONED TO NEXT CLASS.  CLASS VOTED NOT TO 
HAVE THE REVIEW SESSION TUESDAY MORNING.

    2.  Vocabulary from the Grabiner article:  Archimedean axiom

    3.  Improper integrals

        A.  The main example:  Integral of x^{-p} from 1 to 
        infinity.  Work out the 3 cases.

        B.  Basic concepts and pitfalls

            i.  Definitions of CONvergent and DIvergent.  Latter 
            includes both infinite limit and nonexistence of 
            limit.

           ii.  Let G be antiderivative of integrand.  Integral 
            of f to infinity converges iff G has horizontal 
            asymptote.  

          iii.  If f itself has a horizontal asymptote, it must 
            be at 0 if the integral is to converge.  This 
            condition is NOT SUFFICIENT for convergence.
            CLASS EXERCISE:  Sketch f and G for 

                a.  f(x) = x^{-2}

                b.  f(x) = x^{-1/2}
ONE VOLUNTEER TEAM DID WELL ON THIS.

        C.  The other kind of improper integral -- main example:
        Integral of x^{-p} from 0 to 1.
        CLASS EXERCISE:  Fill in the results for the 3 cases.
CLASS HAD SURPRISING DIFFICULTY WITH THIS -- TROUBLE 
UNDERSTANDING THAT 1/h(x) -> infinity if h(x) -> 0.

        D.  Comparison test

            i.  (Quoted from book, p. 493)  0 =< g =< f and
            int^infty f convergent => int^infty g convergent
            (but converse is false).

           ii.  (Not in book; series analog p. 635)
            int^infty |g| convergent => int^infty g convergent
            (if g integrable locally -- see p. 275).

          iii.  Put them together:  |g| =< g and int^infty f
            convergent  => int^infty g convergent.   (needed for 
            next lab)
STUDENTS WERE NOT ATTENTIVE DURING THIS, AND WERE THREATENED WITH 
AN INDIVIDUAL RAT NEXT TIME (IN PLACE OF TEAM EXERCISE ORIGINALLY 
PLANNED).