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In Exercises 21-30use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.

k=1k1+k2.

Short Answer

Expert verified

The seriesk=1k1+k2is convergent.

Step by step solution

01

Step 1. Given information

k=1k1+k2.

02

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be two series with positive terms then,

  1. If limkakbk=L, where Lis any positive real number then either both converge or both diverge.
  2. If limkakbk=0and k=1bkconverges, then k=1akconverges.
  3. Iflimkakbk=andk=1bkdiverges, thenk=1akdiverges.
03

Step 3. The terms of the series ∑k=1∞ k1+k2 are positive.

Find k=1bkfor the given series.

k=1bk=k=1k12k2=k=11k32

04

Step 4. Next find limk→∞ akbk for the given series.

limkakbk=limkk1+k21k32=limkk32k1+k2=limkk21+k2=limkk2k21+1k2=limk11+1k2=1

05

Step 5. From the obtained values,

The value of limkakbk=1which is a finite non zero number.

The value of role="math" localid="1649153106872" k=1bk=k=11k32is convergent by using p-series test.

Therefore, k=1akis convergent.

Hence, the given series is convergent.

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Most popular questions from this chapter

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 qkreturning each year as qk+1=(0.14(1)k+0.36)(qk+h), 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 qe=3hk=10.11k.

(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 qo=6111hk=10.11k.

(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?

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(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" Rn10-6.

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