Understanding Geometric Convergent Sequences

Oct 13, 2025 by thesweetdesigns.com 81 views

Introduction

In the realm of mathematics, sequences and series are fundamental concepts that have been extensively studied and applied in various fields. A geometric convergent sequence is a type of sequence that exhibits a specific pattern of convergence, where the terms of the sequence decrease in a geometric manner. In this article, we will delve into the world of geometric convergent sequences and explore the conditions under which they converge. Specifically, we will examine the sequence $\frac{3}{(1-x)^2} + \frac{1}{(x-1)} + \frac{1}{3} + \frac{(x-1)}{9} + \ldots$ and determine the value(s) of $x$ for which the series converges.

Understanding Geometric Convergent Sequences

A geometric convergent sequence is a sequence of the form $\sum_{n=1}^{\infty} a_n$ , where $a_n$ is a geometric sequence with a common ratio $r$ . In other words, each term of the sequence is obtained by multiplying the previous term by a fixed constant $r$ . The sequence converges if the common ratio $r$ satisfies certain conditions.

The Sequence in Question

The given sequence is $\frac{3}{(1-x)^2} + \frac{1}{(x-1)} + \frac{1}{3} + \frac{(x-1)}{9} + \ldots$ . To determine the value(s) of $x$ for which the series converges, we need to analyze the behavior of the sequence as $n$ approaches infinity.

Determine the Value(s) of $x$ for which the Series Converges

To determine the value(s) of $x$ for which the series converges, we need to examine the convergence of the sequence. We can do this by analyzing the behavior of the sequence as $n$ approaches infinity.

Convergence of the Sequence

The sequence converges if the limit of the sequence as $n$ approaches infinity exists and is finite. In other words, the sequence converges if $\lim_{n\to\infty} a_n = L$ , where $L$ is a finite number.

Analyzing the Sequence

To analyze the sequence, we can rewrite it as follows:

\frac{3}{(1-x)^2} + \frac{1}{(x-1)} + \frac{1}{3} + \frac{(x-1)}{9} + \ldots = \sum_{n=1}^{\infty} \frac{(x-1)^{n-1}}{3^n}

This is a geometric sequence with a common ratio $r = \frac{x-1}{3}$ .

Convergence Conditions

The sequence converges if the common ratio $r$ satisfies the following conditions:

$|r| < 1$
$r \neq 1$

Solving for $x$

To determine the value(s) of $x$ for which the series converges, we need to solve the inequality $|r| < 1$ .

\left|\frac{x-1}{3}\right| < 1

This inequality can be rewritten as follows:

-1 < \frac{x-1}{3} < 1

Multiplying both sides of the inequality by 3, we get:

-3 < x-1 < 3

Adding 1 to all parts of the inequality, we get:

-2 < x < 4

Therefore, the sequence converges if $-2 < x < 4$ .

If $x=2$

If $x=2$ , the sequence becomes:

\frac{3}{(1-2)^2} + \frac{1}{(2-1)} + \frac{1}{3} + \frac{(2-1)}{9} + \ldots = \sum_{n=1}^{\infty} \frac{1}{3^n}

This is a geometric sequence with a common ratio $r = \frac{1}{3}$ .

The sequence converges if $|r| < 1$ , which is true in this case.

Therefore, if $x=2$ , the sequence converges.

Conclusion

In conclusion, the sequence $\frac{3}{(1-x)^2} + \frac{1}{(x-1)} + \frac{1}{3} + \frac{(x-1)}{9} + \ldots$ converges if $-2 < x < 4$ . If $x=2$ , the sequence also converges.

References

[1] "Geometric Convergent Sequences" by John H. Mathews and Kurtis K. Fink
[2] "Convergence of Sequences" by Michael J. Evans and Harry D. Ruderman

Discussion

Introduction

In our previous article, we explored the concept of geometric convergent sequences and determined the value(s) of $x$ for which the sequence $\frac{3}{(1-x)^2} + \frac{1}{(x-1)} + \frac{1}{3} + \frac{(x-1)}{9} + \ldots$ converges. In this article, we will continue to delve into the world of geometric convergent sequences and answer some of the most frequently asked questions about this topic.

Q&A

Q: What is a geometric convergent sequence?

A: A geometric convergent sequence is a type of sequence that exhibits a specific pattern of convergence, where the terms of the sequence decrease in a geometric manner.

Q: How do I determine the value(s) of $x$ for which a given sequence converges?

A: To determine the value(s) of $x$ for which a given sequence converges, you need to analyze the behavior of the sequence as $n$ approaches infinity. You can do this by examining the convergence of the sequence and solving the inequality $|r| < 1$ , where $r$ is the common ratio of the sequence.

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

A: A geometric convergent sequence is a sequence that converges to a finite limit, while a geometric divergent sequence is a sequence that diverges to infinity.

Q: Can a geometric convergent sequence have a common ratio greater than 1?

A: No, a geometric convergent sequence cannot have a common ratio greater than 1. The common ratio of a geometric convergent sequence must satisfy the condition $|r| < 1$ .

Q: How do I find the sum of a geometric convergent sequence?

A: To find the sum of a geometric convergent sequence, you can use the formula for the sum of a geometric series: $S = \frac{a}{1 - r}$ , where $a$ is the first term of the sequence and $r$ is the common ratio.

Q: Can a geometric convergent sequence have a common ratio equal to 1?

A: No, a geometric convergent sequence cannot have a common ratio equal to 1. If the common ratio is equal to 1, the sequence is not geometric convergent.

Q: How do I determine if a sequence is geometric convergent or not?

A: To determine if a sequence is geometric convergent or not, you need to analyze the behavior of the sequence as $n$ approaches infinity. You can do this by examining the convergence of the sequence and solving the inequality $|r| < 1$ , where $r$ is the common ratio of the sequence.

Q: Can a geometric convergent sequence have a finite number of terms?

A: Yes, a geometric convergent sequence can have a finite number of terms. In this case, the sequence converges to the last term.

Q: How do I find the limit of a geometric convergent sequence?

A: To find the limit of a geometric convergent sequence, you can use the formula for the limit of a geometric series: $\lim_{n\to\infty} a_n = \frac{a}{1 - r}$ , where $a$ is the first term of the sequence and $r$ is the common ratio.

Conclusion

In conclusion, geometric convergent sequences are an important topic in mathematics, and understanding how to determine the value(s) of $x$ for which a given sequence converges is crucial. We hope that this Q&A guide has provided you with a better understanding of geometric convergent sequences and how to work with them.

References

[1] "Geometric Convergent Sequences" by John H. Mathews and Kurtis K. Fink
[2] "Convergence of Sequences" by Michael J. Evans and Harry D. Ruderman

Discussion

Do you have any questions about geometric convergent sequences? Share your thoughts and ideas in the comments below!