Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 $$ \sum_{k=1}^{\infty} \frac{2}{(k+2) k} $$

Step-by-Step Solution

Answer

The series converges, and its sum is \( \frac{3}{2} \).

Step 1: Identify the Series

The series given is \( \sum_{k=1}^{\infty} \frac{2}{(k+2) k} \). We need to determine whether this series converges or diverges.

Step 2: Decompose the Fraction

We can decompose the fraction \( \frac{2}{(k+2)k} \) into partial fractions. By using partial fraction decomposition, we find that: \[ \frac{2}{(k+2)k} = \frac{A}{k} + \frac{B}{k+2} \], where \( A \) and \( B \) are constants to be determined.

Step 3: Solve for Constants A and B

Multiply both sides by \((k+2)k \) to clear the denominator: \[ 2 = A(k+2) + Bk \]. Then substitute values: For \( k = 0 \), we get \( A = 1 \). For \( k = -2 \), \( B = -1 \). So, \( \frac{2}{(k+2)k} = \frac{1}{k} - \frac{1}{k+2} \).

Step 4: Recognize as a Telescoping Series

Rewriting the series using the partial fractions gives:\[ \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{k+2} \right) \].This is a telescoping series, where most terms cancel out.

Step 5: Evaluate the Telescopic Cancellation

Write out the first few terms: \( \left( \frac{1}{1} - \frac{1}{3} \right) + \left( \frac{1}{2} - \frac{1}{4} \right) + \left( \frac{1}{3} - \frac{1}{5} \right) + \ldots \).Most terms cancel, leaving: \( 1 + \frac{1}{2} - 0 - 0 \cdots = 1 + \frac{1}{2} = \frac{3}{2} \).

Step 6: Conclusion on Series Behavior

Since the non-cancelled terms yield a finite sum, the series converges. The sum of the series is \( \frac{3}{2} \).

Key Concepts

Telescoping Series

A telescoping series is a special kind of series where many terms cancel each other out. This happens when a series is written such that positive and negative terms eliminate one another as you progress. In simpler terms, think of a telescope that can collapse when pushed together. Similarly, in a telescoping series, the terms collapse.

For example, consider the series:

• \( \sum_{k=1}^{ infinity} \left( \frac{1}{k} - \frac{1}{k+2} \right) \).

If you start writing the first few terms, you will notice:

• \( \left( \frac{1}{1} - \frac{1}{3} \right) + \left( \frac{1}{2} - \frac{1}{4} \right) + \left( \frac{1}{3} - \frac{1}{5} \right) + \ldots \).

Here, the negative part \( \frac{1}{3} \) cancels with the positive part of the next, overlapping term; similarly, others do too. In the end, there are only a few terms left that do not have a partner to cancel with, demonstrating the telescope-like behavior.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler parts. It helps to rewrite a fraction as a sum of simpler fractions. This is particularly useful in series to make them easier to manage.

Given a fraction like \( \frac{2}{(k+2)k} \), partial fraction decomposition allows you to express it as two separate terms:

• \( \frac{2}{(k+2)k} = \frac{1}{k} - \frac{1}{k+2} \).

To find the coefficients for these fractions, you solve the equation:

• \( 2 = A(k+2) + Bk \).

By substituting appropriate values for \( k \) (like 0 and -2), you can determine these constants, which in this exercise was found to be \( A=1 \) and \( B=-1 \). This simplification is crucial for recognizing series as telescoping.

Convergent Series

A convergent series is a series whose terms add up to a finite number. Unlike divergent series that grow infinitely, convergent series settle down to a specific value. Recognizing whether a series converges is vital for understanding its behavior.

With the series \( \sum_{k=1}^{\infty} \frac{1}{k} - \frac{1}{k+2} \), after demonstrating that it behaves as a telescoping series, you can easily spot that the remaining uncancelled terms sum up to a finite value \( \frac{3}{2} \). This finite result implies that the series converges. It's essential to check for convergence to ensure the series doesn't tend toward infinity.

Mathematical Proof

A mathematical proof is the process of demonstrating the truth of a statement using logical reasoning. It's a step-by-step explanation that leads to a particular conclusion, ensuring that each step is based on established principles and rules.

In solving our series problem, the process looked like this:

• First, identify and rewrite the initial complex expression using partial fractions.
• Next, recognize the telescoping nature of the series after decomposition.
• Then evaluate the behavior by checking the cancellations across terms.
• Finally, conclude on convergence based on the remaining terms.

Each of these steps builds upon the previous to justify the claim that the series converges to the sum \( \frac{3}{2} \). This logical method of problem-solving is what constitutes a rigorous mathematical proof.