Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine convergence or divergence for each of the series. Indicate the test you use. \(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\cdots\) Hint: \(a_{n}=\frac{1}{n(n+1)}\)

Step-by-Step Solution

Answer

The series converges by the telescoping series test.

Step 1: Express the General Term

The general term of the series is given by \( a_n = \frac{1}{n(n+1)} \). This will help us determine the behavior of the series as \( n \) goes to infinity.

Step 2: Simplify the General Term

Notice that \( \frac{1}{n(n+1)} \) can be rewritten using partial fractions: \( \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \). This telescopic form is useful for determining convergence.

Step 3: Analyze the Telescoping Series

Write out the terms of the series using the telescoping form: \( \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \cdots \). Most terms cancel out, leaving only the first term of the first fraction and the limit of the last one.

Step 4: Determine Series Convergence

Since the series telescopes to \( \sum_{n=1}^{\infty} (\frac{1}{n} - \frac{1}{n+1}) = 1 \), the series converges. The remaining term after cancellation is 1 as \( n \to \infty \), which indicates the series sums to a finite number.

Key Concepts

Telescoping Series

A telescoping series is a type of infinite series where consecutive terms cancel each other partially or completely. This results in a simplified expression that makes analyzing the series more straightforward.
To see how it works, let's look at the series from the exercise:

• Original series term: \( \frac{1}{n(n+1)} \).
• Transformed into telescoping form: \( \frac{1}{n} - \frac{1}{n+1} \).

This transformation changes how the series looks by creating a situation where each part of a fraction is set to cancel with a part of its subsequent term.
This cancellation drastically simplifies the problem, often leaving just a few terms that don't cancel, helping us quickly find the limit of the series. In this case, the series telescopes to where only the first and a partial piece of the last term remain.

Partial Fraction Decomposition

Partial Fraction Decomposition is an algebraic technique that involves breaking down complex rational expressions into simpler fractions. This method is crucial for simplifying terms in many series, especially those involving products in the denominator.
In our series, the expression \( \frac{1}{n(n+1)} \) can be decomposed as:

• Decomposition: \( \frac{1}{n} - \frac{1}{n+1} \).

This transformation makes it easier to analyze the convergence of the series because it reveals a telescoping pattern. By splitting the fraction this way, each term in the sum from the original series can be expressed in terms that will cancel with one another. This makes complex calculations feasible and opens the door to finding the series' convergence or divergence easily.

Infinite Series Convergence

Infinite series convergence refers to whether or not the sum of all terms in a series approaches a finite value as more terms are added. There are various tests to determine convergence, focusing on different properties or behaviors of series.
For our telescoping series, convergence is determined by:

• Identifying cancellation of intermediate terms.
• Observing the finite result from the remaining terms.

Since the terms in the series sum to converge to a specific number \( (1) \), we conclude it's convergent. The trick was in the cancellation enabled by partial fraction decomposition and the telescoping nature of the series. As a result, despite having an infinite number of terms, the series results in a finite sum through simplification.

Mathematical Series Analysis

Mathematical series analysis involves examining the behavior of series to understand their properties, convergence, divergence, and potential applications. Analyzing a series employs various mathematical tools and tests. Key points to consider:

• Simplification of terms, like partial fraction decomposition.
• Pattern recognition, such as telescoping, to streamline evaluation.
• Utilizing convergence tests to determine if a series sums to a finite number.

For the given exercise, these approaches allowed us to deduce that the series converges to a finite sum. By breaking down complex problems with specific strategies, such as those for converging telescoping series, students can grasp the underlying behavior of series, gaining deeper insight into mathematical patterns and solutions.