Problem 26 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The series is convergent.

Step 1: Examine the General Term

Examine the general term of the series: \( a_n = \frac{n}{n^4 + 1} \). Notice that the degree of the polynomial in the denominator is higher than the degree of the polynomial in the numerator. This typically indicates that \( a_n \) approaches zero as \( n \to \infty \). However, further examination is needed to determine convergence.

Step 2: Simplify the General Term

To assess convergence, we need to consider the behavior of \( a_n = \frac{n}{n^4 + 1} \) for large \( n \). For large \( n \), the general term can be approximated by \( a_n \approx \frac{n}{n^4} = \frac{1}{n^3} \).

Step 3: Apply the Limit Comparison Test

Use the limit comparison test with the series \( \sum \frac{1}{n^3} \), which is a \( p \)-series with \( p = 3 \). \( p \)-series with \( p > 1 \) converge. Compute the limit: \[ \lim_{{n \to \infty}} \frac{\frac{n}{n^4 + 1}}{\frac{1}{n^3}} = \lim_{{n \to \infty}} \frac{n^4}{n^4 + 1} = 1.\]Since this limit is a positive finite number, and \( \sum \frac{1}{n^3} \) converges, \( \sum \frac{n}{n^4 + 1} \) also converges.

Key Concepts

Limit Comparison Test

The limit comparison test is a practical tool in calculus for determining the convergence or divergence of infinite series. You may wonder when to use this test. Typically, you would apply it if the terms of your series involve rational expressions, such as when both the numerator and denominator are polynomials, or when you suspect the terms behave similarly to those of a known convergent or divergent series.

Here’s how it works:

• Choose a series, say \( \sum b_n \), that is known to converge or diverge. Ideally, this should resemble the series you are investigating (for our case, we chose the \( p \)-series \( \sum \frac{1}{n^3} \)).
• Compute the limit \( L \) of the ratio of the terms of the two series: \[ L = \lim_{{n \to \infty}} \frac{a_n}{b_n}. \]
• If \( L \) is a positive finite number, then both series either converge or diverge together.

It's helpful because it connects the behavior of a complex series to a simpler, known series. In our example, since the limit was 1, and the comparison series converges, our original series does too.

P-Series

A \( p \)-series is a fundamental concept in the study of series. It takes the form \( \sum \frac{1}{n^p} \), where \( p \) is a positive real number. The convergent behavior of a \( p \)-series is completely determined by the value of \( p \):

- If \( p > 1 \), the series converges.
- If \( p \leq 1 \), the series diverges.

This rule makes \( p \)-series very useful as a comparison benchmark for other series. In the given solution, the terms of the original series were compared to the terms of the series \( \sum \frac{1}{n^3} \), a \( p \)-series with \( p = 3 \). Because \( p = 3 > 1 \), this series converges. Therefore, since the original series behaves similarly, it also converges.

Asymptotic Behavior

Asymptotic behavior refers to the behavior of functions as the input becomes very large. When analyzing series for convergence, understanding how the terms of the series behave as \( n \to \infty \) is essential.

In the exercise, this involved evaluating the term \( a_n = \frac{n}{n^4 + 1} \). As \( n \) becomes very large, the term simplifies to \( a_n \approx \frac{1}{n^3} \). This simplification is based on recognizing that in the expression \( n^4 + 1 \), the \( n^4 \) term dominates the expression as \( n \to \infty \), making the \( +1 \) negligible by comparison.

This asymptotic analysis simplifies the problem, allowing us to more easily compare the series to standard series like the \( p \)-series, leading to a clearer understanding of its convergence behavior.