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Chapter 7: Q. 9 (page 614)

What is a geometric series? What determines the convergence of a geometric series?

Short Answer

Expert verified

The sum of the terms in geometric progression gives geometric series.

The convergence of the geometric progression depends upon the common ratio.

Step by step solution

01

Step 1. Given information

The series which is obtained by multiplying the terms by a fixed number; the series following is in geometric progression :

a,ar,ar2,....

02

Step 2. Geometric Series

The sum of the terms in geometric progression gives geometric series.

The geometric series is written as 鈭�arkk=0鈭�.

03

Step 3. Convergence of geometric series

The convergence of the geometric progression depends upon the common ratio.

If the common ratio is less than one, the series is convergent.

If the common ratio is greater than one, the series is divergent.

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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(鈭�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=3h鈭�k=1鈭�0.11k.

(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=6111h鈭�k=1鈭�0.11k.

(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?

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=3鈭�鈥�1(k鈭�2)2

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.

3.454545...

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.

Whenever a certain ball is dropped, it always rebounds to a height p% (0 < p < 100) of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of h meters?
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