Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use any test developed so far, including any from Section \(9.2\), to decide about the convergence or divergence of the series. Give a reason for your conclusion. $$ \sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{k+1}\right) $$

Step-by-Step Solution

Answer

The series converges, as it is a telescoping series with a sum of 1.

Step 1: Recognize the Series Type

The given series is \( \sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{k+1}\right) \). We notice that each term can be expressed as a difference of two terms. This indicates that the series is a telescoping series.

Step 2: Expanding the Series

Write out the first few terms of the series: \( \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \ldots \). We observe that adjacent terms cancel each other except for the first term of the first fraction.

Step 3: Use Telescoping Property

Since the series is telescoping, many terms cancel out. The series simplifies to \( \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right) = 1 \). The sequence of partial sums approaches 1 as \( n \to \infty \).

Step 4: Determine Convergence

Since the sequence of partial sums converges to a finite limit (1), the series converges by the definition of convergence of a series.

Key Concepts

Telescoping Series

A telescoping series is a unique type of series characterized by the subtraction of sequential terms, which results in cancellation. In many telescoping series, when you expand the series and add terms together, many intermediate expressions cancel out. This leaves only a few terms of the series, often the first from the initial term and the last from the last term, significantly simplifying the evaluation of the series' sum.

• The structure of a telescoping series makes it easier to identify the convergence or divergence.
• Telescoping series generally take the form: \( \sum_{k=1}^{\infty} (a_k - a_{k+1}) \).
• The large-scale cancellation leads to simpler expressions with clearly evaluated sums.

Telescoping series are valuable in calculus due to this simplification, providing an elegant way to find the sum of infinitely many terms.

Partial Sums

Partial sums are sequences constructed by adding a finite number of terms from a series. They are central to understanding whether a series converges or diverges. For the given telescoping series, each partial sum forms by adding up the terms until the n-th term.

• Partial sums help visualize how a series behaves as it progresses towards infinity.
• The convergence of the series is determined by whether the partial sums approach a finite limit as more terms are added.

In the case of our series, the partial sums simplify dramatically due to cancellation, leading to computations that clarify convergence at each stage.

Infinite Series

An infinite series, like \( \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{k+1} \right) \) expands endlessly as each term of the sequence is added without stopping. Determining whether this sum converges (approaches a finite number) or diverges (grows indefinitely or oscillates without settling) is an essential task in calculus.

• Infinite series are the sum of the terms of a sequence extending to infinity.
• Understanding whether such series converge is crucial for numerous applications in mathematics and physics.

For the series in question, the behavior of its partial sums points towards convergence, showcasing the importance of methods that reveal deeper patterns in such series.

Calculus

Calculus provides the tools and techniques essential for analyzing infinite series and their convergence. It involves the study of limits, derivatives, and integrals, allowing us to deal with change and motion. In the context of series convergence, calculus helps find and prove that an infinite series converges by examining the behavior of its terms and sums.

• Calculus allows for robust convergence tests that assess series convergence or divergence.
• It equips us with methods like the limit of partial sums, which determine if the series approaches a finite value.

Understanding and employing calculus in examining series convergence helps solve problems across various disciplines, including physics, engineering, and economics.