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Chapter 7: Q. 12 (page 603)

Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.

Explain what is important about monotonic and bounded sequences.

Short Answer

Expert verified

If the sequence is decreasing and is bounded below then the sequence is convergent and if the decreasing sequence is not bounded below then the sequence is divergent.

If the sequence is increasing and is bounded above then the sequence is convergent and if the increasing sequence is not bounded below then the sequence is divergent.

Step by step solution

01

Step 1. Given information.

Consider the given question,

The sequence is monotonic and bounded.

02

Step 2. Explain the monotonicity and boundedness of the sequence.

Consider the monotonicity and boundedness of the sequence.

The monotonicity and boundedness of the sequence helps in determining the convergence and divergence of the sequence.

If the sequence is decreasing and is bounded below then the sequence is convergent and if the decreasing sequence is not bounded below then the sequence is divergent.

If the sequence is increasing and is bounded above then the sequence is convergent and if the increasing sequence is not bounded below then the sequence is divergent.

Thus, both the monotonicity and boundedness of the sequence is important for determining the behavior of the sequence.

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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(鈭�1)k+0.36)(qk+h), where h is the number of fish whose spawn she releases from the hatchery annually.

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