Expressing the Series as a Telescoping Sum

Oct 13, 2025 by windailys.com 147 views

Introduction

In mathematics, a series is a sequence of numbers that are added together. The convergence or divergence of a series is a fundamental concept in calculus, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the concept of convergence and divergence of a series, and we will determine whether the given series is convergent or divergent by expressing it as a telescoping sum.

What is a Telescoping Sum?

A telescoping sum is a type of sum that can be expressed as a difference of two terms, where each term is a partial sum of the original series. In other words, a telescoping sum is a sum that can be simplified by canceling out most of the terms, leaving only a few terms. Telescoping sums are useful in determining the convergence or divergence of a series, as they can help us identify the pattern of the series and determine whether it converges or diverges.

The Given Series

The given series is:

\sum_{n=2}^{\infty} \frac{4}{n^2-1}

This series is an infinite series, and it is defined as the sum of the terms $\frac{4}{n^2-1}$ , where $n$ ranges from 2 to infinity.

Expressing the Series as a Telescoping Sum

To determine whether the series is convergent or divergent, we need to express it as a telescoping sum. We can do this by using the following formula:

\frac{4}{n^2-1} = \frac{2}{n-1} - \frac{2}{n+1}

This formula can be verified by multiplying both sides by $(n-1)(n+1)$ , which gives:

4 = 2(n+1) - 2(n-1)

This formula shows that the term $\frac{4}{n^2-1}$ can be expressed as a difference of two terms, $\frac{2}{n-1}$ and $\frac{2}{n+1}$ . This is a telescoping sum, as each term is a partial sum of the original series.

Simplifying the Series

Now that we have expressed the series as a telescoping sum, we can simplify it by canceling out most of the terms. We can do this by writing out the first few terms of the series:

\sum_{n=2}^{\infty} \frac{4}{n^2-1} = \left(\frac{2}{2-1} - \frac{2}{2+1}\right) + \left(\frac{2}{3-1} - \frac{2}{3+1}\right) + \left(\frac{2}{4-1} - \frac{2}{4+1}\right) + \cdots

As we can see, most of the terms cancel out, leaving only a few terms. The series simplifies to:

\sum_{n=2}^{\infty} \frac{4}{n^2-1} = \frac{2}{1} - \frac{2}{\infty}

Determining Convergence or Divergence

Now that we have simplified the series, we can determine whether it converges or diverges. The series converges if the limit of the partial sums exists, and it diverges if the limit of the partial sums does not exist.

In this case, the series converges, as the limit of the partial sums exists. The limit of the partial sums is:

\lim_{n\to\infty} \sum_{k=2}^{n} \frac{4}{k^2-1} = \frac{2}{1} - \frac{2}{\infty} = 2

Conclusion

In conclusion, the series $\sum_{n=2}^{\infty} \frac{4}{n^2-1}$ is convergent, and its sum is 2. We were able to determine this by expressing the series as a telescoping sum and simplifying it. This approach is useful in determining the convergence or divergence of a series, as it can help us identify the pattern of the series and determine whether it converges or diverges.

References

[1] Apostol, T. M. (1974). Calculus. New York: Wiley.
[2] Spivak, M. (1965). Calculus. New York: W.A. Benjamin.

Discussion

The series $\sum_{n=2}^{\infty} \frac{4}{n^2-1}$ is a classic example of a convergent series. It is a telescoping sum, which means that most of the terms cancel out, leaving only a few terms. This approach is useful in determining the convergence or divergence of a series, as it can help us identify the pattern of the series and determine whether it converges or diverges.

However, there are some limitations to this approach. For example, it may not work for all types of series, and it may not be able to determine the convergence or divergence of a series that has a complex pattern.

In addition, there are some other approaches that can be used to determine the convergence or divergence of a series. For example, the ratio test and the root test can be used to determine the convergence or divergence of a series, and they can be more powerful than the telescoping sum approach.

Overall, the telescoping sum approach is a useful tool for determining the convergence or divergence of a series, but it should be used in conjunction with other approaches to ensure that the series is convergent or divergent.

Keywords

Convergence and divergence of series
Telescoping sums
Calculus
Ratio test
Root test
Q&A: Convergence and Divergence of Series =============================================

Introduction

In our previous article, we discussed the concept of convergence and divergence of series, and we used the telescoping sum approach to determine whether the series $\sum_{n=2}^{\infty} \frac{4}{n^2-1}$ is convergent or divergent. In this article, we will answer some frequently asked questions about convergence and divergence of series.

Q: What is the difference between convergence and divergence of a series?

A: Convergence of a series means that the limit of the partial sums exists, and the series approaches a finite value as the number of terms increases. Divergence of a series means that the limit of the partial sums does not exist, and the series does not approach a finite value as the number of terms increases.

Q: How can I determine whether a series is convergent or divergent?

A: There are several methods to determine whether a series is convergent or divergent, including:

Telescoping sum approach: This approach involves expressing the series as a difference of two terms, where each term is a partial sum of the original series.
Ratio test: This test involves calculating the ratio of consecutive terms in the series, and if the ratio is less than 1, the series is convergent.
Root test: This test involves calculating the nth root of the absolute value of the terms in the series, and if the limit of the nth root is less than 1, the series is convergent.
Integral test: This test involves integrating the function that defines the series, and if the integral converges, the series is convergent.

Q: What is a telescoping sum?

A: A telescoping sum is a type of sum that can be expressed as a difference of two terms, where each term is a partial sum of the original series. Telescoping sums are useful in determining the convergence or divergence of a series, as they can help us identify the pattern of the series and determine whether it converges or diverges.

Q: How can I use the ratio test to determine whether a series is convergent or divergent?

A: To use the ratio test, you need to calculate the ratio of consecutive terms in the series. If the ratio is less than 1, the series is convergent. If the ratio is greater than 1, the series is divergent. If the ratio is equal to 1, the test is inconclusive.

Q: How can I use the root test to determine whether a series is convergent or divergent?

A: To use the root test, you need to calculate the nth root of the absolute value of the terms in the series. If the limit of the nth root is less than 1, the series is convergent. If the limit of the nth root is greater than 1, the series is divergent. If the limit of the nth root is equal to 1, the test is inconclusive.

Q: What is the integral test?

A: The integral test is a method for determining whether a series is convergent or divergent. It involves integrating the function that defines the series, and if the integral converges, the series is convergent. If the integral diverges, the series is divergent.

Q: How can I use the integral test to determine whether a series is convergent or divergent?

A: To use the integral test, you need to integrate the function that defines the series. If the integral converges, the series is convergent. If the integral diverges, the series is divergent.

Conclusion

In conclusion, convergence and divergence of series are fundamental concepts in calculus, and there are several methods to determine whether a series is convergent or divergent. The telescoping sum approach, ratio test, root test, and integral test are some of the methods that can be used to determine whether a series is convergent or divergent.

References

[1] Apostol, T. M. (1974). Calculus. New York: Wiley.
[2] Spivak, M. (1965). Calculus. New York: W.A. Benjamin.

Discussion

Convergence and divergence of series are important concepts in calculus, and they have numerous applications in various fields, including physics, engineering, and economics. The telescoping sum approach, ratio test, root test, and integral test are some of the methods that can be used to determine whether a series is convergent or divergent.

However, there are some limitations to these methods. For example, the ratio test and root test may not work for all types of series, and the integral test may not be able to determine the convergence or divergence of a series that has a complex pattern.

In addition, there are some other approaches that can be used to determine the convergence or divergence of a series. For example, the comparison test and the limit comparison test can be used to determine the convergence or divergence of a series by comparing it to a known convergent or divergent series.

Overall, the convergence and divergence of series are important concepts in calculus, and there are several methods that can be used to determine whether a series is convergent or divergent.

Keywords

Convergence and divergence of series
Telescoping sums
Ratio test
Root test
Integral test
Calculus