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Chapter 7: Q. 19 (page 656)
The Alternating Series Test: Let be a sequence of positive numbers. If role="math" localid="1649396442317" for every and , then role="math" localid="1649396524843" .
Let be a sequence of positive numbers. If for every and , then converges .
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Which p-series converge and which diverge?
The contrapositive: What is the contrapositive of the implication 鈥淚f A, then B.鈥�?
Find the contrapositives of the following implications:
If a quadrilateral is a square, then it is a rectangle.
Leila, in her capacity as a population biologist in Idaho, is trying to figure out how many salmon a local hatchery should release annually in order to revitalize the fishery. She knows that ifsalmon spawn in Redfish Lake in a given year, then only fish will return to the lake from the offspring of that run, because of all the dams on the rivers between the sea and the lake. Thus, if she adds the spawn from h fish, from a hatchery, then the number of fish that return from that run k will be .
(a) Show that the sustained number of fish returning approaches as k鈫掆垶.
(b) Evaluate .
(c) How should Leila choose h, the number of hatchery fish to raise in order to hold the number of fish returning in each run at some constant P?
Prove Theorem 7.25. That is, show that the series either both converge or both diverge. In addition, show that if converges to L, thenconverges tolocalid="1652718360109"
Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.
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