Problem 11 - Thomas Calculus

Question

In Exercises \(9-16,\) use the Limit Comparison Test to determine if each series converges or diverges. $$ \sum_{n=2}^{\infty} \frac{n(n+1)}{\left(n^{2}+1\right)(n-1)} $$

Step-by-Step Solution

Answer

The series diverges by the Limit Comparison Test.

Step 1: Simplify the Original Series

Look at the series \( \sum_{n=2}^{\infty} \frac{n(n+1)}{(n^2+1)(n-1)} \). We want to compare this to a known series. Note that the dominant terms in the numerator and denominator as \( n \to \infty \) are \( n^2 \) and \( n^3 \), respectively. Therefore, the expression behaves similarly to \( \frac{n^2}{n^3} = \frac{1}{n} \).

Step 2: Choose a Comparison Series

To use the Limit Comparison Test, choose a series to compare with. Here, we choose the series \( \sum_{n=2}^{\infty} \frac{1}{n} \), which is the harmonic series. We know this series diverges.

Step 3: Apply the Limit Comparison Test

Compute the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \), where \( a_n = \frac{n(n+1)}{(n^2+1)(n-1)} \) and \( b_n = \frac{1}{n} \).Calculate:\[L = \lim_{n \to \infty} \frac{n(n+1) \cdot n}{(n^2+1)(n-1)} = \lim_{n \to \infty} \frac{n^3 + n^2}{n^3 + n^2 - n} = \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{1 + \frac{1}{n} - \frac{1}{n^2}}\]Simplifying:\[L = \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{1 + \frac{1}{n} - \frac{1}{n^2}} = 1 \text{ as } n \to \infty\]Since \( L \) is a positive finite number, the Limit Comparison Test can be applied.

Step 4: Determine Convergence or Divergence

Since \( L \) is finite and positive, and the series \( \sum_{n=2}^{\infty} \frac{1}{n} \) (a harmonic series) diverges, the original series \( \sum_{n=2}^{\infty} \frac{n(n+1)}{(n^2+1)(n-1)} \) also diverges by the Limit Comparison Test.

Key Concepts

Series Convergence

Understanding series convergence is crucial when determining whether an infinite series adds up to a finite value or not. In mathematical terms, a series converges if the sum of its terms eventually approaches a specific number, which is finite. This concept is important in various fields, including physics and engineering, as it helps in solving differential equations and analyzing real-world systems.
For series convergence, there are different tests that can be applied, like the Ratio Test, Root Test, and the Limit Comparison Test, as seen in our original exercise. These tests help us evaluate whether the terms of the series add up to a finite limit or continue indefinitely without settling to a sum.
To apply these tests effectively, you often need to understand the behavior of the terms of the series as they approach infinity. If a series mimics another well-known converging series, it’s likely it converges too. However, if it resembles a known divergent series, like the harmonic series, it may very well diverge.

Harmonic Series

The harmonic series is one of the most famous examples of a divergent series. It is represented as \( \sum_{n=1}^{\infty} \frac{1}{n} \). Despite each term getting smaller and smaller, the series grows without bound, which means it diverges.
The harmonic series is a classical example used in many mathematics courses to illustrate divergence. Attempts to intuitively sum the series lead to divergent results because the sum, as reached by adding all terms, grows larger without converging to a finite number.
In problems like the one we're examining, the harmonic series plays a crucial role in the Limit Comparison Test. By comparing terms of the original series to those of the harmonic series, we can use known properties of the harmonic series (i.e., its divergence) to infer the behavior of the original series.

Dominant Terms in Series

The concept of dominant terms in series refers to identifying which parts of the mathematical expression have the most significant impact on its behavior as \( n \) tends towards infinity. Knowing the dominant terms helps simplify complex expressions and make them manageable for analysis using series tests.
For the series given in the exercise, \( \sum_{n=2}^{\infty} \frac{n(n+1)}{(n^2+1)(n-1)} \), the dominant terms were found to be \( n^2 \) in the numerator and \( n^3 \) in the denominator. These were identified by focusing on the highest power of \( n \) in both the numerator and the denominator as \( n \) becomes very large.
Once dominant terms are identified, they allow us to approximate and compare the original series with simpler ones, like the harmonic series. This process helps in deciding whether the overall behavior of the series converges or diverges as demonstrated with the Limit Comparison Test.