Problem 11 - Thomas Calculus

Question

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=2}^{\infty} \frac{n(n+1)}{\left(n^{2}+1\right)(n-1)}\end{equation}

Step-by-Step Solution

Answer

The series diverges, as confirmed by the Limit Comparison Test with the harmonic series.

Step 1: Identify Comparable Series

The given series is \( \sum_{n=2}^{\infty} \frac{n(n+1)}{(n^2 + 1)(n-1)} \). To apply the Limit Comparison Test, we need to select a simpler series for comparison. A good choice is \( \sum_{n=2}^{\infty} \frac{n^2}{n^3} = \sum_{n=2}^{\infty} \frac{1}{n} \), which is the harmonic series.

Step 2: Express Terms in a Simplified Form

Simplify the terms of the given series for easier comparison. Using leading terms: \( \frac{n(n+1)}{(n^2 + 1)(n-1)} \approx \frac{n^2}{n^3} = \frac{1}{n} \) as \( n \to \infty \). This supports our choice of the harmonic series.

Step 3: Compute the Limit of Ratio of Terms

Apply the Limit Comparison Test, calculating \( \lim_{n \to \infty} \frac{\frac{n(n+1)}{(n^2+1)(n-1)}}{\frac{1}{n}} \). Simplifying the expression, we have \( \lim_{n \to \infty} \frac{n^2+n}{n^3-n} \times n = \lim_{n \to \infty} \frac{n^3+n^2}{n^3-n} \).

Step 4: Evaluate the Limit

Simplify the expression \( \lim_{n \to \infty} \frac{n^3+n^2}{n^3-n} \) by dividing the numerator and the denominator by \( n^3 \), resulting in \( \lim_{n \to \infty} \frac{1+\frac{1}{n}}{1-\frac{1}{n^2}} \). As \( n \to \infty \), this limit approaches \( 1 \).

Step 5: Determine Convergence or Divergence Using the Limit Comparison Test

Since the limit is \( 1 \) and \( 0 < 1 < \infty \), the Limit Comparison Test confirms the behavior of the series is the same as the harmonic series \( \sum \frac{1}{n} \), which diverges. Therefore, the original series \( \sum_{n=2}^{\infty} \frac{n(n+1)}{(n^2+1)(n-1)} \) also diverges.

Key Concepts

Convergence and Divergence

Understanding convergence and divergence is essential in determining the behavior of infinite series. A series converges if the sum of its terms approaches a specific value as more terms are added. This means that as you add terms to the series, the total gets closer to a fixed number, and does not go off to infinity.
A classic example of a convergent series is a geometric series with a common ratio between -1 and 1.
On the other hand, a series diverges if the sum of its terms does not approach a fixed number. Instead, it either increases without bound or doesn't settle on a particular value. An example of a divergent series is the harmonic series, which we will discuss next.
Knowing whether a series converges or diverges helps in various mathematical analyses, including determining the behavior of functions and solving real-world problems.

Harmonic Series

The harmonic series is a fundamental concept in mathematics and can be expressed as \( \ \sum_{n=1}^{\infty} \frac{1}{n} \).
Each term in this series is simply the reciprocal of an integer, and while each term gets smaller, the series as a whole diverges. This means that as you keep adding more terms, the sum keeps increasing without ever approaching a finite limit.

• The harmonic series is one of the simplest examples of a divergent series.
• Despite its divergence, its terms diminish quickly, creating the illusion that it might converge.
• Understanding the divergence of the harmonic series can be useful when analyzing other, more complex series.

The Limit Comparison Test, as used in the original exercise, often relates such complex series back to simpler ones like the harmonic series to determine convergence or divergence.

Series Simplification

Series simplification is a critical step in analyzing series. It involves reducing the complexity of a series to understand its behavior better.
In the original exercise, one series is compared to another, simpler series (the harmonic series) to analyze its behavior. By simplifying the terms of a complex series, you can reveal patterns or distributions akin to those of simpler series.
A common technique in series simplification is to approximate terms. In our example, the expression \( \frac{n(n+1)}{(n^2+1)(n-1)} \) is simplified to \( \frac{1}{n} \) for large \( n \). This aligns it with the harmonic series, making it easier to apply the Limit Comparison Test.
Simplifying series not only aids in computations but also enhances our conceptual understanding, enabling us to apply known results from simple series to more complex ones.