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Chapter 8: Problem 81

What is the difference between a geometric sequence and an infinite geometric series?

Short Answer

Expert verified
The main difference is that a geometric sequence is a list of numbers that follow a pattern (each term is multiplied by a fixed, non-zero number to get the next), whereas an infinite geometric series is the sum of all terms in an infinite geometric sequence. Moreover, only those infinite geometric series have a finite sum where the common ratio is between -1 and 1.

Step by step solution

01

Define a Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number, called the common ratio. For instance, in the sequence 3, 6, 12, 24, ..., the common ratio (r) is 2, and each term can be found by multiplying the previous term by 2.
02

Define an Infinite Geometric Series

An infinite geometric series is the sum of the terms of an infinite geometric sequence. So, considering the sequence from the previous example (3, 6, 12, 24, ...), the corresponding series would be 3 + 6 + 12 + 24 + ... . It's worth mentioning that this series would diverge, meaning it would not have a finite sum, because the common ratio (2) is not between -1 and 1.
03

Differences Between a Geometric Sequence and an Infinite Geometric Series

A geometric sequence and an infinite geometric series differ in the following ways: 1) A geometric sequence is a list of numbers that follow a certain pattern, while an infinite geometric series is the sum of all terms in a geometric sequence. 2) Every geometric sequence can go on indefinitely, whereas not every infinite geometric series has a finite sum. A geometric series only sums to a finite value when the common ratio is between -1 and 1.

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