COALITION MATH 152, SPRING 1998

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

                             Day 29.R


    1.  Vocabulary from the Grabiner article:  Finite differences

    2.  Ratio and root tests:  There are TWO MORE sequences 
    associated with a series -- the ratios |a_{n+1}|/|a_n| and
    the roots \root n \of |a_n|.  If limit of one of these is 
    < 1, the series converges (absolutely).  If limit is > 1, the 
    series diverges.  If limit = 1, the test gives no 
    information.

    3.  Radius of convergence of a power series

        A.  Apply the ratio test to a_n = b_n (x-x_0)^n.

        B.  Conclusion:  Suppose \lim |b_{n+1}/b_n| = 1/R.
        Then series converges if |x-x_0| < R.  Remarks:

            i.  Term "radius" comes from circle in complex plane.

           ii.  Endpoints need to be decided separately (usually 
            a less important issue).

        C.  Examples

            i.  \sum x^n/(3^n n^3)  (R = 3 by either ratio or 
            root test.)

           ii.  tan^{-1} x = \sum (-1)^n x^{2n+1}/(2n+1)
            (Even-order terms are 0.  Must factor out x, regard 
            the rest as a power series in x^2.  R = 1 for x^2, 
            hence for x.)

        D.  More remarks

            i.  Special cases R = \infty (whole plane), R = 0.

           ii.  Even if the limit of |b_{n+1}/|b_n| does not 
           exist, the region of convergence is still a disk.

    4.  Pause for course/instructor evaluations.

    5.  Team RAT:  Fill in the details of the integral-test 
    example on the handout.  (Perform the substitution u = \ln x 
    in the integral \int_1^\infty \ln x /x^p dx, and show the 
    resulting integral converges iff p > 1.)
UNIVERSAL TROUBLE WITH THIS -- I SHOWED THEM HOW TO DO IT.

    6.  Simultaneous challenge:  Prove the assertion of several
    days ago that \sum_{n=1}^\infty 1/2^n = 1.
(THIS CAME UP SOMETIME IN WEEK 28 (BEFORE GEOMETRIC SERIES HAD 
BEEN REEMPHASIZED) IN CONNECTION WITH AN EXPLANATION OF WHY 
POINT 28.T.3.b.iii DOES NOT SAY "IF THE INTEGRAL CONVERGES, THEN 
THE INTEGRAND MUST APPROACH 0".)
ONE TEAM VOLUNTEERED THE ESSENTIAL IDEA BUT NEEDED HELP WITH THE 
DETAILS.

    7.  Return to the list of problems from Sec. 10.7 if time 
    permits.
IT DID NOT.