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Chapter 7: Q. 13 (page 656)

Convergence Tests for Series: Fill in the blanks.

The Divergence Test: If the sequence akdoes not converge to ____, then the series鈭�k=1鈭�ak__.

Short Answer

Expert verified

If the sequence akdoes not converge to 0, then the series 鈭�k=1鈭�akdiverges.

Step by step solution

01

Step 1. Given Information

The given test is the divergence test.

02

Step 2. Explanation

  • According to the divergence test, if the sequence limk鈫�鈭�akdoes not exist or exist but is not equal to 0, then the series 鈭�k=1鈭�akis divergent.
  • So, if akdoes not converge to 0, the series is divergent.

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Most popular questions from this chapter

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�1k2

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?

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder, R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that localid="1649224052075" Rn鈮�10-6.

鈭�k=0鈭�e-k

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.

鈭�k=1鈭�鈥�kek

In Exercises 48鈥�51 find all values of p so that the series converges.

鈭�k=2鈭�1klnkp
See all solutions

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