COALITION MATH 152, SPRING 1998

              (S. A. Fulling, assisted by Vera Rice)

                             Day 28.R


    1.  Vocabulary from the Grabiner article:  Synthetic 
    geometry

    2.  RAT on convergence of improper integrals

        A.  Something that converges at infinity by comparison 
        test.

        B.  Integral from 0 to infinity that converges at both 
        ends, by different comparison tests.

        C.  Something that diverges at infinity by comparison 
        test.
STUDENTS WERE CLEARLY UNPREPARED FOR THIS.  AFTER THE PAPERS WERE 
COLLECTED, WE REDID IT AS A TEAM EXERCISE AND FINALLY GOT SOME 
ATTITUDE ADJUSTMENT AND LEARNING.

    3.   Sequences and series 

        A.  Sequences are discrete analog of functions.

        B.  Series are discrete analog of improper integrals.

        C.  Definition of convergence of a sequence

        D.  Examples of convergent sequences

            i.  Any successful implementation of Newton's method

           ii.  Sequence given by a formula, such as 
                a_n = (n^2 - 1)/(n^2 + 1) 

                a.  The usual methods for limits at infinity 
                (horizontal asymptotes) apply.

                b.  Sequence may converge although the function 
                doesn't, because of oscillation.  (Add
                n^{10} \sin (\pi n) to the formula above.)

          iii.  Sketch graphs of some convergent sequences.  
          Sequence may be nonmonotonic, and may cross over the 
          horizontal asymptote.

        E.  Partial sums are analog of the proper integrals whose 
        limit is the improper integral.  Series are conceptually 
        more elementary than integrals, but usually harder to 
        calculate in closed form.