Which Geometric Series Converges?A. 1 81 + 1 27 + 1 9 + 1 3 + … \frac{1}{81}+\frac{1}{27}+\frac{1}{9}+\frac{1}{3}+\ldots 81 1 ​ + 27 1 ​ + 9 1 ​ + 3 1 ​ + … B. 1 + 1 2 + 1 4 + 1 8 + … 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots 1 + 2 1 ​ + 4 1 ​ + 8 1 ​ + … C. ∑ N = 1 ∞ 7 ( − 4 ) N − 1 \sum_{n=1}^{\infty} 7(-4)^{n-1} ∑ N = 1 ∞ ​ 7 ( − 4 ) N − 1 D. $\sum_{n=1}^{\infty}

Introduction to Geometric Series

A geometric series is a type of infinite series where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. The general form of a geometric series is given by:

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

where $a$ is the first term and $r$ is the common ratio. Geometric series are used to model a wide range of real-world phenomena, including population growth, financial investments, and electrical circuits.

Convergence of Geometric Series

A geometric series converges if and only if the absolute value of the common ratio is less than 1. This is known as the convergence criterion for geometric series. If the common ratio is greater than or equal to 1, the series diverges.

Analyzing the Options

Let's analyze each of the given options to determine which one converges.

Option A: $\frac{1}{81}+\frac{1}{27}+\frac{1}{9}+\frac{1}{3}+\ldots$

The common ratio of this series is $\frac{1}{3}$, which is less than 1. Therefore, this series converges.

Option B: $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots$

The common ratio of this series is $\frac{1}{2}$, which is less than 1. Therefore, this series converges.

Option C: $\sum_{n=1}^{\infty} 7(-4)^{n-1}$

This is a geometric series with first term $a = 7$ and common ratio $r = -4$. The absolute value of the common ratio is $|-4| = 4$, which is greater than 1. Therefore, this series diverges.

Option D: $\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{2^{n-1}}$

This is a geometric series with first term $a = \frac{1}{2}$ and common ratio $r = -\frac{3}{2}$. The absolute value of the common ratio is $|-\frac{3}{2}| = \frac{3}{2}$, which is greater than 1. Therefore, this series diverges.

Conclusion

Based on the analysis, only options A and B converge. The common ratio of option A is $\frac{1}{3}$, and the common ratio of option B is $\frac{1}{2}$. Both of these values are less than 1, which satisfies the convergence criterion for geometric series.

Understanding the Rules

To determine whether a geometric series converges or diverges, we need to check the absolute value of the common ratio. If the absolute value is less than 1, the series converges. If the absolute value is greater than or equal to 1, the series diverges.

Real-World Applications

Geometric series have many real-world applications, including:

• Population growth: A geometric series can be used to model the growth of a population over time.
• Financial investments: A geometric series can be used to calculate the future value of an investment.
• Electrical circuits: A geometric series can be used to model the behavior of electrical circuits.

Conclusion

In conclusion, geometric series are a powerful tool for modeling real-world phenomena. By understanding the rules of convergence, we can determine whether a geometric series converges or diverges. Only options A and B converge, and the common ratio of each series is less than 1.

Final Thoughts

Geometric series are a fundamental concept in mathematics, and understanding their convergence is crucial for many real-world applications. By mastering the rules of convergence, we can unlock the secrets of geometric series and apply them to a wide range of problems.

References

• Kreyszig, E. (2011). Advanced Engineering Mathematics**. John Wiley & Sons._
• Stewart, J. (2016). Calculus: Early Transcendentals**. Cengage Learning._
• Thomas, G. B. (2015). Calculus and Analytic Geometry**. Pearson Education._
  

Introduction

Geometric series are a fundamental concept in mathematics, and understanding their convergence is crucial for many real-world applications. In this article, we will answer some of the most frequently asked questions about geometric series convergence.

Q: What is a geometric series?

A: A geometric series is a type of infinite series where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio.

Q: What is the convergence criterion for geometric series?

A: A geometric series converges if and only if the absolute value of the common ratio is less than 1.

Q: How do I determine whether a geometric series converges or diverges?

A: To determine whether a geometric series converges or diverges, you need to check the absolute value of the common ratio. If the absolute value is less than 1, the series converges. If the absolute value is greater than or equal to 1, the series diverges.

Q: What are some real-world applications of geometric series?

A: Geometric series have many real-world applications, including:

• Population growth: A geometric series can be used to model the growth of a population over time.
• Financial investments: A geometric series can be used to calculate the future value of an investment.
• Electrical circuits: A geometric series can be used to model the behavior of electrical circuits.

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

A: Yes, a geometric series can have a common ratio of 1. However, if the common ratio is 1, the series is not geometric, but rather a constant series.

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

A: Yes, a geometric series can have a common ratio of -1. However, if the common ratio is -1, the series is not geometric, but rather an alternating series.

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

A: To calculate the sum of a convergent geometric series, you can use the formula:

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

where $a$ is the first term and $r$ is the common ratio.

Q: What happens if the common ratio is greater than 1?

A: If the common ratio is greater than 1, the series diverges.

Q: What happens if the common ratio is less than -1?

A: If the common ratio is less than -1, the series diverges.

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

A: No, a geometric series cannot have a common ratio of 0.

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

A: No, a geometric series cannot have a common ratio of infinity.

Conclusion

In conclusion, geometric series are a fundamental concept in mathematics, and understanding their convergence is crucial for many real-world applications. By mastering the rules of convergence, we can unlock the secrets of geometric series and apply them to a wide range of problems.

Final Thoughts

Geometric series are a powerful tool for modeling real-world phenomena. By understanding the rules of convergence, we can determine whether a geometric series converges or diverges. Whether you're a student, a researcher, or a practitioner, geometric series are an essential part of your toolkit.

References

• Kreyszig, E. (2011). Advanced Engineering Mathematics**. John Wiley & Sons._
• Stewart, J. (2016). Calculus: Early Transcendentals**. Cengage Learning._
• Thomas, G. B. (2015). Calculus and Analytic Geometry. Pearson Education.