Day 32 in MAT229

Today:

• Announcements:

  • Your homework for 11.2 is returned: graded
    • #20
    • #49
    • #90

  • Your quiz opportunity tomorrow is over 11.6: Absolute convergence, root and ratio tests

  • You have a homework (11.3) due tomorrow.

• Section 11.4: Comparison Tests

  • Theorems:

    

    Notice how we can use each series to inform us about the other:
    1. if the series of the if finite, so is the smaller series of ;
    2. if the series of the if infinite, so is the larger series of .

    

    This comparison means that we'll need some series in our tool box with which to compare other series. The two obvious candidates are
    1. geometric series, and
    2. p series, including
    3. the harmonic series (to show divergence).
    But once we have a known series, we can throw it into our toolbox, too!

    

    This theorem says that, in the long run, one series has terms which are simply a constant times the terms in the other series.

    In "the long run" means that we only really need to worry about the "tails" of series: we can throw away any finite number of terms for issues of convergence.

  • Examples:
    • #6, p. 750
    • #16
    • #28
    • #34 (draw a picture)
    • #46

• Section 11.5: Alternating Series
  Once again, our mission here is not necessarily the value of the series, but may be simply knowledge of whether it converges or not.

  That being said, Theorem 2 below does give us some help in determining the value of (or at least bounds on the value of) the series of a type called an "alternating" series.

  

  How do they reach the conclusion that the sum is positive and at most ?
  
  • Examples:
    • The alternating harmonic series is convergent (Example 1, p. 753).
    • #5, p. 755
    • #13
    • #17
    • #23
    • #27
    • #34