Properties of Convergent Infinite Geometric Series

Oct 13, 2025 by windailys.com 54 views

Introduction

In mathematics, a geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. An infinite geometric series is a series that has an infinite number of terms, and it is a crucial concept in mathematics, particularly in calculus and algebra. In this article, we will explore the concept of convergent infinite geometric series and determine which statement about it is true.

What is a Convergent Infinite Geometric Series?

A convergent infinite geometric series is a series that has a finite sum, meaning that the sum of the series is a finite number. This occurs when the absolute value of the common ratio, denoted by r, is between -1 and 1, i.e., -1 < |r| < 1. When this condition is met, the series converges to a finite sum, and the sum can be calculated using the formula:

S = a / (1 - r)

where a is the first term of the series, and r is the common ratio.

Statement A: A Finite Geometric Sequence Will Have an Infinite Geometric Series

This statement is false. A finite geometric sequence is a sequence that has a finite number of terms, and it does not have an infinite geometric series. An infinite geometric series, by definition, has an infinite number of terms, and it is not possible for a finite sequence to have an infinite series.

Statement B: An Infinite Geometric Series Converges if -1 < r < 1

This statement is true. As mentioned earlier, an infinite geometric series converges if the absolute value of the common ratio, r, is between -1 and 1, i.e., -1 < |r| < 1. When this condition is met, the series converges to a finite sum, and the sum can be calculated using the formula:

S = a / (1 - r)

Statement C: [Insert Statement C]

Unfortunately, the problem statement does not provide a statement C to evaluate. However, we can discuss the general properties of convergent infinite geometric series and provide examples to illustrate the concept.

Properties of Convergent Infinite Geometric Series

A convergent infinite geometric series has several important properties:

The series converges to a finite sum if and only if the absolute value of the common ratio, r, is between -1 and 1, i.e., -1 < |r| < 1.
The sum of the series can be calculated using the formula: S = a / (1 - r), where a is the first term of the series, and r is the common ratio.
The series converges to a finite sum if and only if the series is absolutely convergent, meaning that the series converges to a finite sum regardless of the order in which the terms are added.

Examples of Convergent Infinite Geometric Series

Here are a few examples of convergent infinite geometric series:

The series 1 + 1/2 + 1/4 + 1/8 + ... converges to a finite sum because the absolute value of the common ratio, r, is between -1 and 1, i.e., -1 < |r| < 1.
The series 2 + 1 + 1/2 + 1/4 + ... converges to a finite sum because the absolute value of the common ratio, r, is between -1 and 1, i.e., -1 < |r| < 1.
The series 1 - 1/2 + 1/4 - 1/8 + ... converges to a finite sum because the absolute value of the common ratio, r, is between -1 and 1, i.e., -1 < |r| < 1.

Conclusion

Q: What is a geometric series?

A: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Q: What is an infinite geometric series?

A: An infinite geometric series is a series that has an infinite number of terms.

Q: What is the condition for an infinite geometric series to converge?

A: An infinite geometric series converges if the absolute value of the common ratio, r, is between -1 and 1, i.e., -1 < |r| < 1.

Q: How do I calculate the sum of a convergent infinite geometric series?

A: The sum of a convergent infinite geometric series can be calculated using the formula: S = a / (1 - r), where a is the first term of the series, and r is the common ratio.

Q: What is the difference between a convergent and a divergent infinite geometric series?

A: A convergent infinite geometric series is a series that has a finite sum, while a divergent infinite geometric series is a series that does not have a finite sum.

Q: Can a finite geometric sequence have an infinite geometric series?

A: No, a finite geometric sequence cannot have an infinite geometric series. A finite geometric sequence is a sequence that has a finite number of terms, while an infinite geometric series is a series that has an infinite number of terms.

Q: What are some examples of convergent infinite geometric series?

A: Some examples of convergent infinite geometric series include:

The series 1 + 1/2 + 1/4 + 1/8 + ...
The series 2 + 1 + 1/2 + 1/4 + ...
The series 1 - 1/2 + 1/4 - 1/8 + ...

Q: What are some examples of divergent infinite geometric series?

A: Some examples of divergent infinite geometric series include:

The series 1 + 2 + 4 + 8 + ...
The series 1 - 2 + 4 - 8 + ...
The series 1 + 1 + 1 + 1 + ...

Q: Can I use the formula S = a / (1 - r) to calculate the sum of a divergent infinite geometric series?

A: No, you cannot use the formula S = a / (1 - r) to calculate the sum of a divergent infinite geometric series. This formula is only valid for convergent infinite geometric series.

Q: What is the relationship between convergent infinite geometric series and absolute convergence?

A: A convergent infinite geometric series is absolutely convergent, meaning that the series converges to a finite sum regardless of the order in which the terms are added.

Q: Can I use the properties of convergent infinite geometric series to solve problems involving infinite series?

A: Yes, you can use the properties of convergent infinite geometric series to solve problems involving infinite series. The properties of convergent infinite geometric series can be applied to a wide range of problems involving infinite series.

Conclusion

In conclusion, we have discussed the properties of convergent infinite geometric series and provided examples to illustrate the concept. We have also answered some frequently asked questions about convergent infinite geometric series.