Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=2}^{\infty}\left(\frac{1}{k}-\frac{1}{k-1}\right) $$

Step-by-Step Solution

Answer

The series converges to -1.

Step 1: Understand the Series

The given series is \( \sum_{k=2}^{\infty} \left( \frac{1}{k} - \frac{1}{k-1} \right) \). This is a telescoping series, meaning that each term cancels out part of the adjacent terms.

Step 2: Write out the First Few Terms

Writing the first few terms of the series helps in identifying the pattern:\(\left( \frac{1}{2} - \frac{1}{1} \right) + \left( \frac{1}{3} - \frac{1}{2} \right) + \left( \frac{1}{4} - \frac{1}{3} \right) + \ldots\)You can see that each \( -\frac{1}{k-1} \) should eventually cancel with \( \frac{1}{k-1} \) of the next term.

Step 3: Identify the Telescoping Pattern

Notice that in the telescoping series, most terms cancel out:\(-1 + 0 + 0 + \ldots + \frac{1}{n} - \frac{1}{n}\)The only terms that do not get cancelled out are \( -1 \) and \( \frac{1}{n} \) as \( n \to \infty \). Thus, the series simplifies to \(-1 + \frac{1}{n} \).

Step 4: Determine Convergence and Sum

As \( n \to \infty \), the term \( \frac{1}{n} \to 0 \). Therefore, the sum of the series converges to \(-1\). The series is telescoping and converges to \(-1\).

Key Concepts

Telescoping Series

When dealing with a telescoping series, we are looking at a type of infinite series where a pattern of terms cancel each other out, significantly simplifying the problem. This is often seen in series where each term is separated by addition or subtraction. Writing out the initial few terms can reveal how terms cancel out, similar to how the series provided, \(\sum_{k=2}^{\infty} \left( \frac{1}{k} - \frac{1}{k-1} \right)\), simplifies. Here, for example, the \(-\frac{1}{2}\) of one term cancels out the \(\frac{1}{2}\) of the next.This cancellation phenomenon leads to only a few terms remaining. These leftover terms then help determine the series' sum easily. While telescoping, remember, it's crucial to explicitly state which terms cancel and which remain as the series progresses. This process is pivotal in analyzing and understanding series' convergence.

Infinite Series

An infinite series is a sum of infinitely many terms. Mathematically, it can be represented as \(a_1 + a_2 + a_3 + \ldots\), where each \(a_i\) is a term in the series. Unlike finite sums, an infinite series continues indefinitely but could still converge to a specific value.To comprehend infinite series, we often rely on finding a pattern within the terms and determining whether this leads towards any particular value. Such series are central in calculus and other mathematical fields due to their fascinating properties and practical applications.In the context of the telescoping series, understanding its infinite nature helps us see why it converges despite extending to infinity. Essentially, even series with no observable endpoint can summarize to a finite sum.

Mathematical Convergence

Mathematical convergence refers to the behavior of an infinite series as it progresses towards a finite limit. The series converges if adding an infinite number of terms results in a sum that approaches a specific number.One reliable method to check for convergence is to evaluate the limit of the partial sums of the series as they extend to infinity. If these partial sums settle on a particular value, the series is said to converge to that point.In the example of our telescoping series, by writing out and simplifying the terms, the process leads to understanding why the series tends towards \(-1\) as \(n\) becomes infinitely large. Recognizing this trend helps not just in proving mathematical convergence but also in calculating the exact sum for series that meet these criteria. Understanding convergence is crucial for gauging whether an infinite series has a meaningful and calculable result.