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Chapter 7: Q. 12 (page 631)

Let 0<p<1. Show that 0鈮�1klnk鈮�1kpfor large values of k. Explain why we cannot use a p-series with0<p<1in a comparison test to verify the divergence of the series鈭�k=2鈭�1klnk.

Short Answer

Expert verified

The comparison test provides no information on the series鈭�k=2鈭�1k(lnk) divergence.

Step by step solution

01

Step 1. Given information

An expression is given as0鈮�1klnk鈮�1kp

02

Step 2. Verification

For larger values of k ,

kp鈮�kkPlnk鈮�klnk1klnk鈮�1kplnk1kplnk<1kp1klnk<1kp

For 0<p<1, the p-series is divergent. If the series 鈭�k=1xbkconverges, then the series 鈭�i=1鈭�akconverges, according to the comparison test.

But the series 鈭�i=1鈭�bk=鈭�k=1鈭�1kpis divergent, so we cannot find the value of convergence.

The comparison test provides no information on the series' divergence.

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Most popular questions from this chapter

Explain why a function a(x) has to be continuous in order for us to use the integral test to analyze a series 鈭�k=1鈭�akfor convergence.

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.

鈭�k=1鈭�1k

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 ifpksalmon spawn in Redfish Lake in a given year, then only 0.2pkfish 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 pk+1=0.2(pk+h)..

(a) Show that the sustained number of fish returning approaches p鈭�=h鈭�k+1鈭�0.2kas k鈫掆垶.

(b) Evaluate p鈭�.

(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?

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

鈭�k=1鈭�鈥�k鈭�3/2

Use either the divergence test or the integral test to determine whether the series in Exercises 32鈥�43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

35.鈭�k=1鈭�k2+k2
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