Problem 19 - Calculus: Early Transcendentals

Question

Determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} \frac {n^3}{n^4 + 4} \)

Step-by-Step Solution

Answer

The series diverges.

Step 1: Determine the Behavior of the Series

To determine if the series \( \sum_{n=1}^{\infty} \frac{n^3}{n^4 + 4} \) converges or diverges, we need to analyze the behavior of its general term as \( n \to \infty \). Simplifying \( \frac{n^3}{n^4 + 4} \) for large \( n \), we recognize that it behaves similarly to \( \frac{n^3}{n^4} = \frac{1}{n} \).

Step 2: Use the Limit Comparison Test

For the limit comparison test, compare \( a_n = \frac{n^3}{n^4 + 4} \) with \( b_n = \frac{1}{n} \). Calculate the limit: \[ \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{n^3}{n^4 + 4}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n^4}{n^4 + 4} = \lim_{n \to \infty} \frac{1}{1 + \frac{4}{n^4}} = 1. \] Since the limit is a positive finite number, the series \( \sum a_n \) and \( \sum b_n \) either both converge or both diverge.

Step 3: Determine the Convergence of the Comparison Series

The series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is the harmonic series, which is known to diverge. This implies that the series \( \sum_{n=1}^{\infty} \frac{n^3}{n^4 + 4} \) also diverges by the limit comparison test.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool in calculus for determining the convergence or divergence of a series. It involves comparing the series in question with another series whose convergence behavior is already understood. By doing so, we can infer the behavior of the original series without the need for complex calculations.
The idea behind this test is simple: given two series, if the ratio of their terms approaches a positive finite limit as the number of terms goes to infinity, then both series either converge or diverge together.

• Let’s consider two series \( \sum a_n \) and \( \sum b_n \).
• The test states: if the limit \(lim_{n \to \infty} \frac{a_n}{b_n} = c\) where \(c\) is a positive finite number, then the series \(\sum a_n\) and \(\sum b_n\) must behave in the same way.

In practical terms, when we have a series like \(\sum \frac{n^3}{n^4 + 4}\), we compare it to a simpler, known series \(\sum \frac{1}{n}\), allowing us to leverage our understanding of well-studied series to make judgments about less familiar ones.

Convergence and Divergence

In the study of infinite series, determining whether a series converges or diverges is crucial.
**Convergence** means that as more terms are added, the series approaches a specific value, while **divergence** indicates that the series does not settle to any particular value.
To determine the behavior of a series, mathematicians employ various tests, with the Limit Comparison Test being just one among many.

• A series \(\sum a_n\) converges if the sequence of partial sums \(S_N = a_1 + a_2 + \ldots + a_N\) approaches a limit as \(N\to\infty\).
• Otherwise, the series diverges.

In practical terms, analyzing a series's convergence allows us to discern valuable properties about functions and sequences, contributing to fields like physics, engineering, and finance. The initial step usually involves examining the behavior of series terms as they move towards infinity. In the problem above, once we establish this relationship using a series like the harmonic series, we see that the target series diverges.

Harmonic Series

The harmonic series is a cornerstone concept in understanding series behavior, especially divergence. Its significance primarily stems from its straightforward form and its divergent nature, despite terms decreasing slowly.
The harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\) can be written as the sum \(1 + \frac{1}{2} + \frac{1}{3} + \ldots\).
Despite each term becoming smaller, the series itself is known to diverge.

• This shows the intriguing nature of series: a series can have terms nearing zero but still diverge.
• The divergence of the harmonic series is a vital reference point in calculus, often used as a benchmark for applying tests like the Limit Comparison Test.

Understanding this concept is pivotal, since such a simple form of a series, by itself, underpins much of the theoretical groundwork for inferring behaviors of more complex series.