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Chapter 7: Q. 91 (page 593)
Prove that a sequence that is both increasing and decreasing is constant.
Proved
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Determine whether the series converges or diverges. Give the sum of the convergent series.
Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A divergent series in which .
(b) A divergent p-series.
(c) A convergent p-series.
Given a series , in general the divergence test is inconclusive when . For a geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.
Ifconverges, explain why we cannot draw any conclusions about the behavior of.
Leila finds that there are more factors affecting the number of salmon that return to Redfish Lake than the dams: There are good years and bad years. These happen at random, but they are more or less cyclical, so she models the number of fish returning each year as , where h is the number of fish whose spawn she releases from the hatchery annually.
(a) Show that the sustained number of fish returning in even-numbered years approach approximately
(Hint: Make a new recurrence by using two steps of the one given.)
(b) Show that the sustained number of fish returning in odd-numbered years approaches approximately
(c) How should Leila choose h, the number of hatchery fish to breed in order to hold the minimum number of fish returning in each run near some constant P?
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