Question

Writing In Exercises 89 and 90 , use a graphing utility to determine the first term that is less than 0.0001 in each of the convergent series. Note that the answers are very different. Explain how this will affect the rate at which each series converges. $$ \sum_{n=1}^{\infty} \frac{1}{n(n+1)} $$

Short answer

By following the step-by-step solution, you will iteratively compute the terms of the series until you find the term that first falls below 0.0001 and make observations about the rate of series convergence. The exact term number will depend on the specific computations.

Step-by-step solution

Understand the Problem

The series under consideration is given by \( \sum_{n=1}^{\infty} \frac{1}{n(n+1)} \). Our task is to find the first term in this series that is less than 0.0001.

Compute the terms

Calculate the terms of the series iteratively, using the formula \( \frac{1}{n(n+1)} \), where n is the term number. Start with n=1 and increase n by 1 for each subsequent term. Continue this until getting a term less than 0.0001.

Determine the First Term Less Than 0.0001

Keep track of the term number when the term value first falls below 0.0001. This term number is the required solution.

Analyze the series convergence

The difference between each term and the next in a series represents the rate at which the series converges. If the term less than 0.0001 is far along in the sequence, that might indicate slower convergence, while an earlier appearance indicates faster convergence.

Key concepts

Graphing Utility

When studying mathematics, especially series, a graphing utility is a valuable tool that enables us to visualize the behavior of mathematical functions and series. By plotting the terms of a series, one can quickly determine patterns, behavior at infinity, and even guess the series' convergence or divergence.

For the exercise in question, a graphing utility can be used to plot the terms of the series \( \frac{1}{n(n+1)} \) for increasing values of n. As you proceed through the series, the plotted points decrease in value. By observing when these points drop below a certain threshold—in this case, 0.0001—we can identify the first term meeting the criterion without manually calculating each term. This visual approach can reveal the answer more quickly and intuitively than by calculation alone.

It's crucial to remember that while graphing utilities offer an excellent way to comprehend series, they might not always give the precision required for certain mathematical proof or understanding the nuanced behavior at extreme values. However, they serve as a bridge to grasp the concept of convergence visually.

Series Convergence

The concept of series convergence is central in understanding how a series behaves as its terms are added indefinitely. A series is said to be convergent if the sum of its infinite terms approaches a finite number—contrarily, it's divergent if the sum goes to infinity, or does not settle on a specific value.

For the series \( \frac{1}{n(n+1)} \) presented in the exercise, the convergence is guaranteed by the fact that the terms are positive and steadily decreasing to zero. This particular type is known as a telescoping series, where successive terms cancel out each other partly, making it easier to evaluate their sum.

Understanding whether a series converges, and if so, how quickly, is essential not just in pure mathematics, but also in fields that rely on the summing of series or approximations, such as physics, engineering, and quantitative finance. Determining convergence can involve tests such as the ratio test, root test, integral test, or by understanding the nature of the series like the p-series or geometric series. By mastering these, students can dissect complex problems into understandable segments.

Rate of Convergence

The rate of convergence is a measure of how quickly a series approaches its limit. For a convergent series, understanding the rate of convergence helps in estimating how many terms we need to include to achieve a certain degree of accuracy.

The exercise provided centers around this idea, seeking the first term less than 0.0001, essentially pinpointing the threshold of accuracy. When we find that term, we can analyse the progression towards it. If the terms decrease rapidly, the series has a fast rate of convergence, and we require fewer terms to come close to the series' limit. Conversely, if the terms decrease slowly, the rate of convergence is slow, and we might need many terms to reach the same level of accuracy.

For the given series \( \frac{1}{n(n+1)} \) the rate of convergence will affect how quickly we can reach an approximation of the sum to within 0.0001 of the actual value. It demonstrates the practical importance of the rate of convergence, not just in a mathematical context, but in any computational simulation or model where we replace an infinite series with a finite sum. By evaluating this rate, we can implement better algorithms for calculations that are not just efficient but also reliable for various applications.