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Chapter 7: Q. 20 (page 604)
State the converse of Theorem 7.19. Explain why Theorem 7.20 is a partial converse.
The converse of the theorem is that if the sequence is convergent then the sequence is bounded and monotonic.
But the converse of the theorem is not true.
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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 7.24 (a). That is, show that if c is a real number and is a convergent series, then .
Which p-series converge and which diverge?
Let be any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to . (Hint: Argue that if you add up some finite number of the terms of , the sum will be greater than . Then argue that, by adding in some other finite number of the terms of
, you can get the sum to be less than . By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to .)
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.
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