Problem 13 - Thomas Calculus

Question

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{n}{n+1} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{n}{n+1} \) diverges by the Divergence Test.

Step 1: Identify the Series Form

The given series is \( \sum_{n=1}^{\infty} \frac{n}{n+1} \). This is an infinite series with each term given by \( a_n = \frac{n}{n+1} \).

Step 2: Determine Limit of Terms

To determine convergence, find if \( \lim_{{n \to \infty}} a_n \) is zero. Evaluate, \( \lim_{{n \to \infty}} \frac{n}{n+1} = \lim_{{n \to \infty}} \frac{n}{n+1} = \lim_{{n \to \infty}} \frac{1}{1+\frac{1}{n}} = 1 \). Since the limit is 1 and not zero, this suggests that the series may not converge.

Step 3: Apply the Divergence Test

Use the Divergence Test, which states that if the limit of \( a_n \) as \( n \to \infty \) is not zero, the series diverges. Since \( \lim_{{n \to \infty}} \frac{n}{n+1} = 1 eq 0 \), by the Divergence Test, the series diverges.

Key Concepts

Divergence Test

The Divergence Test is a simple yet powerful tool in determining whether an infinite series converges or diverges. It primarily assesses the limit of the sequence of terms in the series. The basic idea behind the Divergence Test relies on the behavior of these terms as they progress towards infinity.
An infinite series, represented as \( \sum a_n \), will only converge if the sequence of individual terms \( a_n \) approaches zero. This means the limit of the individual terms as \( n \to \infty \) must be zero, i.e., \( \lim_{{n \to \infty}} a_n = 0 \).

• If \( \lim_{{n \to \infty}} a_n eq 0 \), then the series \( \sum a_n \) strongly diverges. This is because accumulation of non-zero terms will not sum to a finite number.
• If \( \lim_{{n \to \infty}} a_n = 0 \), the Divergence Test is inconclusive. In such cases, other methods must be used to evaluate convergence or divergence.

Remember, the Divergence Test is a necessary condition for convergence but not sufficient. For our series \( \sum_{n=1}^{\infty} \frac{n}{n+1} \), each term's limit is 1 (not zero), hence the series diverges.

Infinite Series

An infinite series is the sum of infinitely many terms, usually expressed in the form \( \sum_{n=1}^{\infty} a_n \). These series are a foundational concept in calculus and mathematical analysis, particularly when exploring limits and convergence.
The sum of the series is often aimed to be evaluated to see if it converges to a definite value or diverges.

• Convergence: Occurs when the infinite sum approaches a specific finite number as more terms are added.
• Divergence: Happens when the sum does not settle on a finite value, instead either increasing indefinitely or oscillating.

The exploration of infinite series involves different tests, of which the Divergence Test is just one. Understanding infinite series helps in various applications, from solving differential equations to representing functions as sums.

Limit of a Sequence

The concept of the limit of a sequence is integral to understanding series and their convergence. A sequence is simply a list of numbers written in a particular order, often denoted as \( a_1, a_2, a_3, \ldots \). As we consider more terms in a sequence, we get closer to understanding its behavior towards infinity.
To find the limit of a sequence \( \{a_n\} \), examine whether \( a_n \) approaches a specific number \( L \) as \( n \) becomes very large. Mathematically, this is noted as \( \lim_{{n \to \infty}} a_n = L \).

• If \( L \) is a finite number, the sequence converges to \( L \).
• If the terms do not approach a specific number, the sequence is divergent.

The evaluation of this limit is a key step in performing the Divergence Test on an infinite series. In our series \( \sum_{n=1}^{\infty} \frac{n}{n+1} \), the limit of the sequence \( \frac{n}{n+1} \) is 1. Because the limit is not zero, it indicates that the series diverges.