Understanding the Telescoping Series and the Value of an Infinite Sum

In mathematical analysis, the concept of a series is fundamental to understanding various phenomena. One intriguing type of series is the telescoping series, which can simplify the sum of an infinite series significantly. This article delves into one such example, the sum of the series that involves fractional terms and alternating signs.

The Given Series

The problem at hand involves the summation of the series:

( S sum_{n1}^{infty} frac{left(-1right)^{n-1}}{2n-12n-3} )

Breaking Down the Series

To understand this series completely, we can break it down using partial fractions. The given series can be rewritten as:

( S frac{1}{4} sum_{n1}^{infty} -1^{n-1} left( frac{1}{2n-1} - frac{1}{2n-3} right) )

Exploring the Telescoping Effect

The telescoping series property allows us to see how most intermediate terms cancel out. Let's see this in action with the partial sum up to (N):

( S_N frac{1}{4} sum_{n1}^{N} -1^{n-1} left( frac{1}{2n-1} - frac{1}{2n-3} right) frac{1}{4} left( frac{-1^{1-1}}{2cdot1-1} - frac{-1^{N-1-2-1}}{2N-1-3} - frac{-1^{N-2-1}}{2N-2-3} right) )

As (N) approaches infinity, most of the terms inside the sum will cancel each other out, leaving only the first and last terms. Since the remaining terms tend to zero via the Squeeze Theorem, we effectively ignore their contributions.

Calculating the Limit

By taking the limit as (N) approaches infinity, we are left with the following:

( S frac{1}{4} left( 1 - frac{1}{3} - 0 - 0 right) frac{1}{6} )

Conclusion

Thus, the value of the given infinite series is ( frac{1}{6} ). This example beautifully illustrates the power of telescoping series and how they can be used to simplify seemingly complex sums.

Frequently Asked Questions (FAQ)

What is a telescoping series? A telescoping series is a series where the terms cancel out, leaving only a few remaining terms that are easy to sum. How is the series simplified? The series is simplified by expressing each term as the difference of two fractions, which allows the intermediate terms to cancel out. Why is the Squeeze Theorem used here? The Squeeze Theorem is used to show that the remaining terms tend to zero, allowing us to ignore their contributions to the sum.

References

Wikipedia: Telescoping Series

Main Points

The given series is a telescoping series. Partial fractions are used to break down the series into simpler components. The Squeeze Theorem helps to determine the limit of the remaining terms. The final value of the series is ( frac{1}{6} ).