Given The Series $ -\frac{4}{7} + \frac{4}{49} - \frac{4}{343} + \frac{4}{2401} \cdots }$Does This Series Converge Or Diverge?- Diverges- ConvergesIf The Series Converges, Find The Sum { \square$ $

Introduction

Alternating series are a type of infinite series where the terms alternate between positive and negative values. These series are commonly used in mathematics to model real-world phenomena, such as the behavior of electrical circuits or the growth of populations. In this article, we will examine the convergence of the given alternating series: ${-\frac{4}{7} + \frac{4}{49} - \frac{4}{343} + \frac{4}{2401} \cdots}$. We will determine whether this series converges or diverges and, if it converges, find its sum.

Understanding Alternating Series

Alternating series have the general form: $a_1 - a_2 + a_3 - a_4 + \cdots}$. The key characteristic of an alternating series is that the terms alternate between positive and negative values. In the given series, the terms are ${-\frac{4{7}, \frac{4}{49}, -\frac{4}{343}, \frac{4}{2401}, \cdots}$. We can see that the terms alternate between positive and negative values.

The Alternating Series Test

The alternating series test is a mathematical tool used to determine whether an alternating series converges or diverges. The test states that if an alternating series satisfies the following conditions, then it converges:

1. The terms of the series decrease in absolute value, i.e., ${|a_{n+1}| \leq |a_n|}$
2. The limit of the terms as $n$ approaches infinity is zero, i.e., ${\lim_{n\to\infty} a_n = 0}$

Applying the Alternating Series Test

Let's apply the alternating series test to the given series. We can see that the terms decrease in absolute value, i.e., ${|-\frac{4}{7}| > |\frac{4}{49}| > |-\frac{4}{343}| > |\frac{4}{2401}| > \cdots}$. Additionally, the limit of the terms as $n$ approaches infinity is zero, i.e., ${\lim_{n\to\infty} \frac{4}{7 \cdot 7^n} = 0}$. Therefore, the given series satisfies the conditions of the alternating series test and converges.

Finding the Sum of the Series

Now that we have established that the series converges, we can find its sum. The sum of an alternating series can be found using the formula: $S = a_1 - a_2 + a_3 - a_4 + \cdots = \sum_{n=1}^{\infty} (-1)^{n+1} a_n}$. In this case, the sum is ${S = -\frac{47} + \frac{4}{49} - \frac{4}{343} + \frac{4}{2401} \cdots}$. We can use the formula for the sum of an infinite geometric series to find the sum ${S = \frac{a_1{1 - r}}$, where $a_1$ is the first term and $r$ is the common ratio. In this case, $a_1 = -\frac{4}{7}$ and $r = -\frac{1}{7}$.

Calculating the Sum

Now that we have the formula for the sum, we can calculate the sum: ${S = \frac{-\frac{4}{7}}{1 - (-\frac{1}{7})} = \frac{-\frac{4}{7}}{\frac{8}{7}} = -\frac{4}{8} = -\frac{1}{2}}$. Therefore, the sum of the series is ${-\frac{1}{2}}$.

Conclusion

In conclusion, the given alternating series converges and its sum is ${-\frac{1}{2}}$. The alternating series test was used to determine whether the series converges or diverges, and the formula for the sum of an infinite geometric series was used to find the sum. This analysis demonstrates the importance of the alternating series test in determining the convergence of alternating series and finding their sums.

References

• Krantz, S. G. (2013). "Calculus: Early Transcendentals". McGraw-Hill Education.
• Stewart, J. (2015). "Calculus: Early Transcendentals". Cengage Learning.
• Thomas, G. B. (2015). "Calculus and Analytic Geometry". Pearson Education.

Further Reading

For further reading on alternating series and their applications, we recommend the following resources:

• "Alternating Series" by Wolfram MathWorld
• "Alternating Series Test" by Math Open Reference
• "Alternating Series" by Khan Academy
  

Q: What is an alternating series?

A: An alternating series is a type of infinite series where the terms alternate between positive and negative values. The general form of an alternating series is: ${a_1 - a_2 + a_3 - a_4 + \cdots}$.

Q: What is the alternating series test?

A: The alternating series test is a mathematical tool used to determine whether an alternating series converges or diverges. The test states that if an alternating series satisfies the following conditions, then it converges:

1. The terms of the series decrease in absolute value, i.e., ${|a_{n+1}| \leq |a_n|}$
2. The limit of the terms as $n$ approaches infinity is zero, i.e., ${\lim_{n\to\infty} a_n = 0}$

Q: How do I determine whether an alternating series converges or diverges?

A: To determine whether an alternating series converges or diverges, you can use the alternating series test. Check if the terms of the series decrease in absolute value and if the limit of the terms as $n$ approaches infinity is zero.

Q: What is the sum of an alternating series?

A: The sum of an alternating series can be found using the formula: $S = a_1 - a_2 + a_3 - a_4 + \cdots = \sum_{n=1}^{\infty} (-1)^{n+1} a_n}$. In some cases, the sum can be found using the formula for the sum of an infinite geometric series ${S = \frac{a_1{1 - r}}$, where $a_1$ is the first term and $r$ is the common ratio.

Q: Can an alternating series have a finite sum?

A: Yes, an alternating series can have a finite sum. For example, the series ${-\frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} \cdots}$ has a finite sum of ${-\frac{1}{2}}$.

Q: Can an alternating series have an infinite sum?

A: Yes, an alternating series can have an infinite sum. For example, the series ${-\frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} - \cdots}$ has an infinite sum.

Q: What are some common applications of alternating series?

A: Alternating series are commonly used in mathematics to model real-world phenomena, such as the behavior of electrical circuits or the growth of populations. They are also used in physics to describe the behavior of oscillating systems.

Q: How do I find the sum of an alternating series with a common ratio?

A: To find the sum of an alternating series with a common ratio, you can use the formula: ${S = \frac{a_1}{1 - r}}$, where $a_1$ is the first term and $r$ is the common ratio.

Q: Can I use the alternating series test to determine whether a series converges or diverges if the terms do not decrease in absolute value?

A: No, the alternating series test requires that the terms decrease in absolute value. If the terms do not decrease in absolute value, you cannot use the alternating series test to determine whether the series converges or diverges.

Q: What are some common mistakes to avoid when working with alternating series?

A: Some common mistakes to avoid when working with alternating series include:

• Assuming that an alternating series converges or diverges based on the behavior of the first few terms.
• Failing to check if the terms decrease in absolute value.
• Using the alternating series test incorrectly.

Q: How do I determine whether an alternating series is absolutely convergent or conditionally convergent?

A: To determine whether an alternating series is absolutely convergent or conditionally convergent, you can use the ratio test or the root test to check if the series converges absolutely. If the series converges absolutely, it is absolutely convergent. If the series converges but not absolutely, it is conditionally convergent.