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Chapter 12: Q 100 (page 827)

Makeup two infinite geometric series, one that has a sum and one that does not. Give them to a friend and ask for the sum of each series.

Short Answer

Expert verified

Infinite geometric series that has a sum is 1+13+19+127鈥�

Infinite geometric series that has no sum is3+6+12+24+...

Step by step solution

01

Step 1. Given Information

Makeup two infinite geometric series, one that has a sum and one that does not.

02

Step 2. Infinite geometric series that has a sum

An infinite geometric series has a sum if and only if the common ratio of the corresponding geometric sequence is between 0 and 1 .

Therefore if a geometric series involves a common ratio of 13, then it has a sum

Let's choose 1 as the first term for the geometric series.

One infinite geometric series that has a sum is 1+13+19+127鈥�

Sum of the geometric series is11-13=123=32

03

Step 3. Infinite geometric series that has no sum 

An infinite geometric series has no sum when the common ratio is not between 0 and 1

Let the common ratio be 2 and the first term be 3, then the infinite sequence has no sum

3+6+12+24+...

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Most popular questions from this chapter

Problems 17鈥�28, write down the first five terms of each sequence.

cn=-1n+1n2

In Problems 37鈥�50, a sequence is defined recursively. Write down the first five terms.

a1=-2;an=n+an-1

In Problems 11鈥�16, evaluate each factorial expression.

5!8!3!

In Problems 29鈥�36, the given pattern continues. Write down the nth term of a sequence ansuggested by the pattern.

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In Problems 37鈥�50, a sequence is defined recursively. Write down the first five terms.

a1=5;an=2an-1
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