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if 0ak1kfor every positive integer k, then the series k=1akconverges. The objective is to whether determine the statement is true or false

Short Answer

Expert verified

false

Step by step solution

01

To determine whether given statement is true or false 

To determine whether the given statement is true or not by using comparison test

02

According to comparison test

k=1akand k=1bkare two series with positive terms such that 0akbkfor every positive integer k

If k=1bkconverges then the series k=1akalso converges

If the series k=1bk=k=11kis divergent by the power series, So k=1akalso divergence

Hence the given statement is wrong.

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Most popular questions from this chapter

For each series in Exercises 44–47, 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 thatRn10-6

k=11k2

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.

k=0ekk=0ek

Determine whether the series k=0-98kconverges or diverges. Give the sum of the convergent series.

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?

Find the values of x for which the seriesk=0xkconverges.

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