Problem 14 - 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. $$ \sum_{k=6}^{\infty} \frac{2}{k-5} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Write the series in a recognizable form

The series given is \( \sum_{k=6}^{\infty} \frac{2}{k-5} \). We can rewrite this series as \( \sum_{n=1}^{\infty} \frac{2}{n} \) by substituting \( n = k - 5 \). This substitution simplifies the series to a harmonic series, \( 2 \sum_{n=1}^{\infty} \frac{1}{n} \).

Step 2: Identify the type of series

The series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a harmonic series, which is known to diverge. A harmonic series is the sum of terms in the form \( \frac{1}{n} \).

Step 3: Determine convergence or divergence

Since the series resembles the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), and the harmonic series is known to diverge, our series \( 2\sum_{n=1}^{\infty} \frac{1}{n} \) also diverges. Multiplying a divergent series by a constant does not change its divergence.

Key Concepts

Harmonic Series

The concept of a harmonic series is a fundamental one in mathematical analysis. It is a type of infinite series where each term decreases in a harmonic manner. More specifically, the harmonic series is the summation of the reciprocals of natural numbers, given by \[ \sum_{n=1}^{ \infty} \frac{1}{n} \]When we dive deeper into its properties, we find the harmonic series diverges. This means that as you sum more and more terms, the total does not settle at a particular number. It keeps growing beyond any limits. This might seem counterintuitive because each term is getting smaller, but the series just swells infinitely.
For example, writing out the first few terms looks like:

• \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots\)

Unlike finite sums, harmonic series trickle towards infinity, despite each individual step getting tinier. Understanding the divergence of this sequence is crucial in recognizing how some seemingly convergent series may actually behave.

Infinite Series

An infinite series is a way to add up infinitely many numbers in a structured mode. You can think of it as adding terms that go on without an end. Each infinite series is made of terms that are summed up sequentially.
Here, the context is quite important. Infinite series can either converge or diverge, meaning they might settle to a number or grow endlessly. Convergence happens when the partial sums get closer and closer to a finite limit. On the other hand, divergence occurs when they refuse to stabilize. If you visualize an infinite series as a long train, the cars just keep attaching, forming a never-ending chain. Some chains, like geometric series with certain properties, take you to a specific spot. Harmonic series and some others, keep you on a perpetual journey. Being able to identify whether your series converges or diverges is key in calculus and helps predict the nature of complicated sums determined by functions, analyses, or real-world problems. Recognizing patterns, such as those in harmonic series, can simplify complex situations immensely.

Mathematical Substitution

Substitution is a mathematical strategy used to simplify complex problems. It involves replacing expressions with variables to handle problems in neat and manageable ways.
In the given task, substitution came into play by letting \( n \) be the replacement for \( k - 5 \). This transformation adjusted the series from \(\sum_{k=6}^{\infty} \frac{2}{k-5} \) to the recognizable harmonic series \(2\sum_{n=1}^{\infty} \frac{1}{n} \).Substitution offers numerous benefits:

• It reduces clutter and makes the structure of a problem clearer.
• The approach is often used for integrals and series to make known patterns more visible.
• By making substitutions, you align with patterns you've previously understood, like the properties of harmonic series.

All these tactics help in both simplifying the process logically and guiding towards the right conclusion about convergence or divergence. Use substitution cleverly, and it often opens the door to straightforward solutions.