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A question about Upper Darboux Integrals

  1. Dec 5, 2012 #1
    In this link:

    http://math.berkeley.edu/~scanez/courses/math104/fall11/homework/hw10-solns.pdf

    For qustion 32.6, I'm not sure if I'm understanding how there can be a "sequence" of upper and lower darboux integrals.

    So (for example), what is the difference between [tex]U_{10}[/tex] and [tex]U_{11}[/tex]? Does it mean that [tex]U_{10}[/tex] is the upper darboux integral when there are 10 partitions (10 rectangles)...and [tex]U_{11}[/tex] is the upper darboux integral when there are 11 partitions? So, if we have [tex]U_1[/tex], does it mean that there is only one partition (so only one rectangle)?

    Thanks in advance
     
  2. jcsd
  3. Dec 5, 2012 #2
    No, for each n, U_n is just any upper Darboux sum. It corresponds to some partition P_n which could have any number of subintervals. Likewise L_n is any old sequence of lower Darboux sums which corresponds to some other partition Q_n. The hypothesis is that the difference (U_n-L_n) converges to 0.

    There is no stipulated relation between consecutive partitions in the sequence. In particular they are not assumed to be refinements of previous partitions.
     
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