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Chapter 7: Q. 10 (page 624)

What is meant by the remainder Rn of a series ∑k=1∞ak

Short Answer

Expert verified

The remainder of the series ∑k=1∞ak is Rn=∑k=n+1∞ak.

Step by step solution

01

Step 1. Given Information.

The series:

∑k=1∞ak

02

Step 2. The remainder of the convergent series.

If ∑k=1∞ak is a convergent series with sum L, then we may approximate L with the nth partial sum Sn=∑k=1nak. The nth remainder is defined by

Rn=L-Sn=∑k=n+1∞ak

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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=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?

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

∑k=1∞1k

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?

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.

0.6345345...

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=2∞ 1kln⁡k
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