Expressing the Series as a Geometric Series

Oct 13, 2025 by wbrum.org 87 views

Introduction

Infinite series are a fundamental concept in mathematics, and solving them can be a challenging task. In this article, we will explore the solution to the infinite series $1 - \frac{6}{7} + \frac{36}{49} - \cdots + (-1)^{38} \frac{6^{38}}{7^{38}}$ . We will break down the problem into smaller parts, analyze the pattern, and use mathematical techniques to find the sum of the series.

Understanding the Series

The given series is an alternating series, where each term is obtained by multiplying the previous term by a constant ratio. The series starts with 1, and each subsequent term is obtained by multiplying the previous term by $-\frac{6}{7}$ .

1 - \frac{6}{7} + \frac{36}{49} - \cdots + (-1)^{38} \frac{6^{38}}{7^{38}}

Identifying the Pattern

To solve the series, we need to identify the pattern and find a way to express it in a more manageable form. Let's analyze the first few terms of the series:

$1 = \frac{7}{7}$
$-\frac{6}{7} = -\frac{6}{7} \cdot \frac{7}{7}$
$\frac{36}{49} = \frac{6^2}{7^2}$
$-\frac{216}{343} = -\frac{6^3}{7^3}$

We can see that each term is obtained by multiplying the previous term by $-\frac{6}{7}$ , and the numerator is a power of 6, while the denominator is a power of 7.

Expressing the Series as a Geometric Series

Now that we have identified the pattern, we can express the series as a geometric series. A geometric series is a series of the form:

a + ar + ar^2 + \cdots + ar^{n-1}

where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

In our case, the first term is 1, the common ratio is $-\frac{6}{7}$ , and the number of terms is infinite. We can express the series as:

1 - \frac{6}{7} + \frac{36}{49} - \cdots + (-1)^{38} \frac{6^{38}}{7^{38}} = \sum_{n=0}^{\infty} (-1)^n \left(\frac{6}{7}\right)^n

Using the Formula for the Sum of an Infinite Geometric Series

The sum of an infinite geometric series is given by the formula:

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

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

In our case, the first term is 1, and the common ratio is $-\frac{6}{7}$ . Plugging these values into the formula, we get:

S = \frac{1}{1 - \left(-\frac{6}{7}\right)} = \frac{1}{1 + \frac{6}{7}} = \frac{1}{\frac{13}{7}} = \frac{7}{13}

Conclusion

In this article, we solved the infinite series $1 - \frac{6}{7} + \frac{36}{49} - \cdots + (-1)^{38} \frac{6^{38}}{7^{38}}$ using mathematical techniques. We identified the pattern of the series, expressed it as a geometric series, and used the formula for the sum of an infinite geometric series to find the sum.

Final Answer

$The final answer is $\boxed{\frac{7}{13}}$.<br/> **Solving the Infinite Series: A Q&A Guide**$

Introduction

In our previous article, we solved the infinite series $1 - \frac{6}{7} + \frac{36}{49} - \cdots + (-1)^{38} \frac{6^{38}}{7^{38}}$ using mathematical techniques. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is an infinite series?

A: An infinite series is a sum of an infinite number of terms. It is a mathematical expression that represents the sum of an infinite number of values.

Q: What is a geometric series?

A: A geometric series is a series of the form:

a + ar + ar^2 + \cdots + ar^{n-1}

where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How do I identify the pattern of an infinite series?

A: To identify the pattern of an infinite series, you need to analyze the first few terms of the series and look for a common ratio or a pattern in the terms.

Q: How do I express an infinite series as a geometric series?

A: To express an infinite series as a geometric series, you need to identify the first term and the common ratio of the series. You can then use the formula for the sum of an infinite geometric series to find the sum.

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 = \frac{a}{1 - r}

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

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 the first term and the common ratio into the formula. You can then simplify the expression to find the sum.

Q: What is the sum of the infinite series $1 - \frac{6}{7} + \frac{36}{49} - \cdots + (-1)^{38} \frac{6^{38}}{7^{38}}$ ?

A: The sum of the infinite series $1 - \frac{6}{7} + \frac{36}{49} - \cdots + (-1)^{38} \frac{6^{38}}{7^{38}}$ is $\boxed{\frac{7}{13}}$ .

Q: How do I apply the solution to real-world problems?

A: The solution to the infinite series can be applied to real-world problems in various fields such as finance, engineering, and economics. For example, the solution can be used to model population growth, compound interest, and other phenomena that involve infinite series.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the infinite series $1 - \frac{6}{7} + \frac{36}{49} - \cdots + (-1)^{38} \frac{6^{38}}{7^{38}}$ . We hope that this guide has been helpful in answering your questions and providing a deeper understanding of the solution.

Final Answer

The final answer is $\boxed{\frac{7}{13}}$ .