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Chapter 7: Q. 10 (page 655)

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.

The remainder of a convergent series

Short Answer

Expert verified

The difference between the nth partial sum and the sum of a series is the remainder of the convergent series.

For example, in the sequence,12+14+18+116+132+...the remainder of the convergent series isR1=14+18+116+132+....,R2=18+116,,,,Rn=12n+1+12n+2+12n+3+...

Step by step solution

01

Step 1. Given Information    

The given statement is the remainder of a convergent series

02

Step 2. Explanation    

The difference between the nth partial sum and the sum of a series is the remainder of the convergent series.

For example, consider a sequence 12+14+18+116+..

The first few partial sums and remainders are as follows,

S1=12,R1=14+18+116+...S2=12+14,R2=18+116+132+...

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Most popular questions from this chapter

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.

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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(−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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