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Chapter 7: Q. 90 (page 593)
Prove that the ratio of successive terms of a nonzero geometric sequence is constant
Proved
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If a positive finite number, what may we conclude about the two series?
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?
Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.
Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.
Let f(x) be a function that is continuous, positive, and decreasing on the interval such that , What can the integral tells us about the series ?
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