Convergence and Divergence of Infinite Geometric Series

Oct 13, 2025 by windailys.com 2 views

Introduction

In mathematics, an infinite geometric series is a type of series that consists of an infinite number of terms, where each term is obtained by multiplying the previous term by a fixed constant. The series is said to converge if the sum of the terms approaches a finite limit, and it is said to diverge if the sum of the terms approaches infinity. In this article, we will discuss the concept of infinite geometric series and determine which of the given series diverge.

What is an Infinite Geometric Series?

An infinite geometric series is a series of the form:

a + ar + ar^2 + ar^3 + \ldots

where $a$ is the first term and $r$ is the common ratio. The series is said to converge if $|r| < 1$ , and it is said to diverge if $|r| \geq 1$ .

Convergence and Divergence of Infinite Geometric Series

The convergence or divergence of an infinite geometric series can be determined using the following formula:

S = \frac{a}{1 - r}

where $S$ is the sum of the series. If $|r| < 1$ , the series converges to a finite limit, and if $|r| \geq 1$ , the series diverges.

Divergence of Infinite Geometric Series

A series diverges if the common ratio $r$ is greater than or equal to 1. In this case, the series does not approach a finite limit, and the sum of the terms approaches infinity.

Example 1: Series A

Let's consider the series:

27 + 9 + 3 + 1 + \ldots

This series can be written as:

27 + 27\left(\frac{1}{3}\right) + 27\left(\frac{1}{3}\right)^2 + 27\left(\frac{1}{3}\right)^3 + \ldots

The common ratio of this series is $r = \frac{1}{3}$ . Since $|r| < 1$ , the series converges.

Example 2: Series B

Let's consider the series:

1 + \frac{4}{3} + \frac{16}{9} + \frac{64}{27} + \ldots

This series can be written as:

1 + 1\left(\frac{4}{3}\right) + 1\left(\frac{4}{3}\right)^2 + 1\left(\frac{4}{3}\right)^3 + \ldots

The common ratio of this series is $r = \frac{4}{3}$ . Since $|r| > 1$ , the series diverges.

Example 3: Series C

Let's consider the series:

2.2 + 0.22 + 0.022 + \ldots

This series can be written as:

2.2 + 2.2\left(\frac{1}{10}\right) + 2.2\left(\frac{1}{10}\right)^2 + 2.2\left(\frac{1}{10}\right)^3 + \ldots

The common ratio of this series is $r = \frac{1}{10}$ . Since $|r| < 1$ , the series converges.

Example 4: Series D

Let's consider the series:

\sum_{n=1}^{\infty} \frac{2}{3^n}

This series can be written as:

\frac{2}{3} + \frac{2}{3^2} + \frac{2}{3^3} + \frac{2}{3^4} + \ldots

The common ratio of this series is $r = \frac{1}{3}$ . Since $|r| < 1$ , the series converges.

Conclusion

In conclusion, an infinite geometric series is a type of series that consists of an infinite number of terms, where each term is obtained by multiplying the previous term by a fixed constant. The series is said to converge if the sum of the terms approaches a finite limit, and it is said to diverge if the sum of the terms approaches infinity. We have discussed the concept of infinite geometric series and determined which of the given series diverge.

References

[1] "Infinite Geometric Series" by Math Open Reference
[2] "Convergence and Divergence of Infinite Geometric Series" by Khan Academy
[3] "Infinite Geometric Series" by Wolfram MathWorld

Introduction
What is an Infinite Geometric Series?
Convergence and Divergence of Infinite Geometric Series
Divergence of Infinite Geometric Series
Example 1: Series A
Example 2: Series B
Example 3: Series C
Example 4: Series D
Conclusion
References
Table of Contents
Infinite Geometric Series: Q&A =====================================

Q: What is an infinite geometric series?

A: An infinite geometric series is a type of series that consists of an infinite number of terms, where each term is obtained by multiplying the previous term by a fixed constant.

Q: How do I determine if an infinite geometric series converges or diverges?

A: To determine if an infinite geometric series converges or diverges, you need to find the common ratio (r) of the series. If |r| < 1, the series converges. If |r| ≥ 1, the series diverges.

Q: What is the formula for the sum of an infinite geometric series?

A: The formula for the sum of an infinite geometric series is:

S = a / (1 - r)

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

Q: Can an infinite geometric series have a negative common ratio?

A: Yes, an infinite geometric series can have a negative common ratio. However, the series will still converge if |r| < 1.

Q: Can an infinite geometric series have a common ratio of 1?

A: No, an infinite geometric series cannot have a common ratio of 1. If r = 1, the series will diverge.

Q: Can an infinite geometric series have a common ratio of -1?

A: No, an infinite geometric series cannot have a common ratio of -1. If r = -1, the series will diverge.

Q: How do I find the common ratio of an infinite geometric series?

A: To find the common ratio of an infinite geometric series, you need to divide each term by the previous term. For example, if the series is 1 + 2 + 4 + 8 + ..., you can find the common ratio by dividing each term by the previous term: 2/1 = 2, 4/2 = 2, 8/4 = 2. Therefore, the common ratio is 2.

Q: Can an infinite geometric series have a common ratio of 0?

A: No, an infinite geometric series cannot have a common ratio of 0. If r = 0, the series will converge to the first term.

Q: Can an infinite geometric series have a common ratio of infinity?

A: No, an infinite geometric series cannot have a common ratio of infinity. If r = infinity, the series will diverge.

Q: How do I use the formula for the sum of an infinite geometric series?

A: To use the formula for the sum of an infinite geometric series, you need to plug in the values of a and r into the formula: S = a / (1 - r). For example, if the series is 1 + 2 + 4 + 8 + ..., a = 1 and r = 2. Plugging these values into the formula, you get: S = 1 / (1 - 2) = 1 / (-1) = -1.

Q: Can an infinite geometric series have a sum of infinity?

A: Yes, an infinite geometric series can have a sum of infinity. This occurs when the common ratio is greater than or equal to 1.

Q: Can an infinite geometric series have a sum of 0?

A: Yes, an infinite geometric series can have a sum of 0. This occurs when the common ratio is 0.

Q: Can an infinite geometric series have a sum of a negative number?

A: Yes, an infinite geometric series can have a sum of a negative number. This occurs when the common ratio is negative and |r| < 1.

Q: Can an infinite geometric series have a sum of a fraction?

A: Yes, an infinite geometric series can have a sum of a fraction. This occurs when the common ratio is a fraction and |r| < 1.

Q: Can an infinite geometric series have a sum of a decimal?

A: Yes, an infinite geometric series can have a sum of a decimal. This occurs when the common ratio is a decimal and |r| < 1.

Q: Can an infinite geometric series have a sum of a negative decimal?

A: Yes, an infinite geometric series can have a sum of a negative decimal. This occurs when the common ratio is a negative decimal and |r| < 1.

Q: Can an infinite geometric series have a sum of a fraction with a negative sign?

A: Yes, an infinite geometric series can have a sum of a fraction with a negative sign. This occurs when the common ratio is a negative fraction and |r| < 1.

Q: Can an infinite geometric series have a sum of a decimal with a negative sign?

A: Yes, an infinite geometric series can have a sum of a decimal with a negative sign. This occurs when the common ratio is a negative decimal and |r| < 1.

Q: Can an infinite geometric series have a sum of a complex number?

A: Yes, an infinite geometric series can have a sum of a complex number. This occurs when the common ratio is a complex number and |r| < 1.

Q: Can an infinite geometric series have a sum of a complex number with a negative sign?

A: Yes, an infinite geometric series can have a sum of a complex number with a negative sign. This occurs when the common ratio is a negative complex number and |r| < 1.

Q: Can an infinite geometric series have a sum of a complex number with a fraction?

A: Yes, an infinite geometric series can have a sum of a complex number with a fraction. This occurs when the common ratio is a complex number with a fraction and |r| < 1.

Q: Can an infinite geometric series have a sum of a complex number with a decimal?

A: Yes, an infinite geometric series can have a sum of a complex number with a decimal. This occurs when the common ratio is a complex number with a decimal and |r| < 1.

Q: Can an infinite geometric series have a sum of a complex number with a negative sign and a fraction?

A: Yes, an infinite geometric series can have a sum of a complex number with a negative sign and a fraction. This occurs when the common ratio is a negative complex number with a fraction and |r| < 1.

Q: Can an infinite geometric series have a sum of a complex number with a negative sign and a decimal?

A: Yes, an infinite geometric series can have a sum of a complex number with a negative sign and a decimal. This occurs when the common ratio is a negative complex number with a decimal and |r| < 1.

Q: Can an infinite geometric series have a sum of a complex number with a fraction and a decimal?

A: Yes, an infinite geometric series can have a sum of a complex number with a fraction and a decimal. This occurs when the common ratio is a complex number with a fraction and a decimal and |r| < 1.

Q: Can an infinite geometric series have a sum of a complex number with a negative sign, a fraction, and a decimal?

A: Yes, an infinite geometric series can have a sum of a complex number with a negative sign, a fraction, and a decimal. This occurs when the common ratio is a negative complex number with a fraction and a decimal and |r| < 1.

Conclusion

References

[1] "Infinite Geometric Series" by Math Open Reference
[2] "Convergence and Divergence of Infinite Geometric Series" by Khan Academy
[3] "Infinite Geometric Series" by Wolfram MathWorld

Introduction
What is an Infinite Geometric Series?
Convergence and Divergence of Infinite Geometric Series
Divergence of Infinite Geometric Series
Example 1: Series A
Example 2: Series B
Example 3: Series C
Example 4: Series D
Q&A
Conclusion
References
Table of Contents