Problem 29 - Thomas Calculus

Question

In Exercises \(27-34,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$ \sum_{n=0}^{\infty} \frac{1}{n+4} $$

Step-by-Step Solution

Answer

The n-th Term Test is inconclusive as the limit of the n-th term is 0.

Step 1: Understand the n-th Term Test for Divergence

The n-th Term Test for Divergence states that if the limit of the n-th term of a series as \(n\) approaches infinity is not zero, then the series diverges. If the limit is zero, the test is inconclusive.

Step 2: Identify the n-th Term of the Given Series

The given series is \(\sum_{n=0}^{\infty} \frac{1}{n+4}\). The n-th term \(a_n\) is \(\frac{1}{n+4}\).

Step 3: Calculate the Limit of the n-th Term

Calculate \(\lim_{{n \to \infty}} \frac{1}{n+4}\). As \(n\) approaches infinity, \(n+4\) also approaches infinity, therefore \(\frac{1}{n+4}\) approaches zero.

Step 4: Interpret the Result

Since \(\lim_{{n \to \infty}} \frac{1}{n+4} = 0\), the n-th Term Test for Divergence is inconclusive in this case. This means that the series may either converge or diverge, but this test cannot determine which.

Key Concepts

Limits of Sequences

When we talk about the limit of a sequence, we are exploring what happens to the terms of that sequence as the number of those terms increases indefinitely. In simpler terms, the sequence \( a_n \) has a limit \( L \) if the values of \( a_n \) get closer and closer to \( L \) as \( n \) becomes very large. Limiting behavior is foundational to understanding how sequences behave over time.

In mathematical notation, we express the limit of a sequence \( a_n \) as \( L \) by writing \[\lim_{{n \to \infty}} a_n = L.\]Some key points about the limits of sequences include:

• If the limit exists and is a finite number, the sequence converges to that limit.
• If the limit does not exist, or is infinite, the sequence diverges.
• Understanding limits helps us predict the "end behavior" of sequences.

Therefore, limits serve as a tool to help us analyze whether a series of numbers will get infinitely large, shrink infinitely small, or stabilize at a particular value as they extend to infinity.

Divergent Series

A series is a sum of sequences, and when we consider whether a series is divergent, we are asking if adding its terms together in an endless process leads to an infinitely large value. If a series is divergent, it means the sum does not settle into a fixed number. Understanding divergence is essential because some series may appear to have sums that fluctuate wildly, much like divergent series tend to do.

The nth-Term Test for Divergence is one established method to check for divergence. It highlights:

• If the limit of the nth term of a series \( a_n \) as \( n \to \infty \) is not zero, then the series is divergent.
• If the limit is zero, the test becomes inconclusive, meaning the series may either converge or diverge.

The series given in the example, \( \sum_{n=0}^{\infty} \frac{1}{n+4} \), showed an inconclusive result with this test because its limit is zero. This means alternative methods are required to ascertain if the series is truly divergent or convergent.

Convergent Series

In contrast to divergent series, a convergent series is one whose terms, when summed, lead to a finite total. This means as you add more and more terms from the sequence, the overall sum "settles" into a particular value rather than increasing indefinitely or fluctuating without bound.

Key features of convergent series include:

• The sum approaches a specific real number, even as more terms are added.
• The series does not exhibit endless growth or chaotic behavior.
• Convergence demands that terms decrease in size quickly enough to reach a finite sum.

For instance, the geometric series and the harmonic series are traditional examples where one might observe convergent behaviors, albeit in vastly different circumstances. Understanding convergence helps in determining the stability and outcomes of series in mathematical analysis and applications. Though the nth-Term Test couldn't conclude the convergence of our example series, understanding convergence in broader contexts remains critical for deeper analytical studies.