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Chapter 7: Q. 13 (page 624)

Explain why a function a(x) has to be continuous in order for us to use the integral test to analyze a series ∑k=1∞akfor convergence.

Short Answer

Expert verified

The function a(x)has to be continuous because if the function is not continuous, the improper integral ∫1∞a(x)dxmay not be defined.

So if the integral is not defined, the convergence or divergence cannot be examined.

Step by step solution

01

Step 1. Given Information.

The function: a(x)

The series:

∑k=1∞a(k)

02

Step 2. The integral test.

If f(x):[1,∞)→ℝis continuous, eventually positive and decreasing on [1,∞), and fkis the sequence defined byfk={f(k)} for every k∈ℤ+ , then∑k=1∞fkand∫1∞f(x)dx either both converge or diverge.

03

Step 3. Why it has to be continuous.

The function a(x)has to be continuous because if the function is not continuous, the improper integral may not be defined.

So if the integral is not defined, the convergence or divergence cannot be examined.

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Most popular questions from this chapter

Given thata0=-3,a1=5,a2=-4,a3=2and∑akk=2∞=7, find the value of∑akk=4∞.

Prove Theorem 7.24 (a). That is, show that if c is a real number and∑k=1∞ak is a convergent series, then ∑k=1∞cak=c∑k=1∞ak.

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=3h∑k=1∞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∑k=1∞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?

Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.

0.237237237...

An Improper Integral and Infinite Series: Sketch the function f(x)=1xfor x ≥ 1 together with the graph of the terms of the series ∑k=1∞1k.Argue that for every term Snof the sequence of partial sums for this series,Sn>∫1n+11xdx. What does this result tell you about the convergence of the series?
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