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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 sum of a convergent series

Short Answer

Expert verified

A series converges if its sequence of partial sums converges, and in that case, we define the sum of the series to be the limit of its partial sums. For example, n=112n is the series which converges.

Step by step solution

01

Step 1. Given Information   

The given statement is the sum of a convergent series

02

Step 2. Explanation     

A series converges if its sequence of partial sums converges, and in that case, we define the sum of the series to be the limit of its partial sums.

The sum of the first terms of a series up to a point is another sequence called the partial sum. A series is convergent if its partial sum has a limit. We can say that a convergent series is equal to some finite value.

For example, n=112nis a convergent series as the partial sums will look like 12,34,78,..and converges to 1.

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Most popular questions from this chapter

Determine whether the series k=04k+132kconverges or diverges. Give the sum of the convergent series.

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

Find the values of x for which the series k=03xkconverges.

Determine whether the series k=0πe3kconverges 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?

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