Problem 9 - Thomas Calculus

Question

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\begin{array}{l}{\sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3}} \\ {\text { (Hint: Limit Comparison with } \sum_{n=1}^{\infty}\left(1 / n^{2}\right) )}\end{array}\end{equation}

Step-by-Step Solution

Answer

The series converges.

Step 1: Identify the Given Series

The given series is \( \sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3} \). Our task is to determine its convergence or divergence using the Limit Comparison Test.

Step 2: Select a Comparison Series

From the hint, we select the comparison series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), which is a well-known convergent p-series with \( p=2 > 1 \).

Step 3: Evaluate the Limit for the Limit Comparison Test

We compute the limit \( \lim_{n \to \infty} \frac{\frac{n-2}{n^{3}-n^{2}+3}}{\frac{1}{n^2}} \). Simplifying, this becomes \( \lim_{n \to \infty} \frac{n^2(n-2)}{n^3-n^2+3} \).

Step 4: Simplify the Limit Expression

Simplify \( \lim_{n \to \infty} \frac{n^2(n-2)}{n^3-n^2+3} \) to \( \lim_{n \to \infty} \frac{n^3 - 2n^2}{n^3 - n^2 + 3} \). Divide the numerator and the denominator by \( n^3 \), resulting in \( \lim_{n \to \infty} \frac{1 - \frac{2}{n}}{1 - \frac{1}{n} + \frac{3}{n^3}} \).

Step 5: Calculate the Limit

Taking the limit as \( n \to \infty \), \( \frac{1 - \frac{2}{n}}{1 - \frac{1}{n} + \frac{3}{n^3}} \to \frac{1}{1} = 1 \).

Step 6: Apply the Limit Comparison Test Conclusion

Since the limit is a positive finite constant (1), by the Limit Comparison Test, the series \( \sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3} \) converges, just like the comparison series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \).

Key Concepts

Convergence

Convergence in the context of series is a fundamental concept in calculus. It refers to the behavior of an infinite series as you sum its terms. When we say that a series converges, we mean that as you add more and more terms, the total sum approaches a specific number. This limiting value is known as the "limit" of the series. Convergent series are significant because they give you a finite result, making them quite useful in various mathematical and real-world applications.

For instance, the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is an example of a convergent series. Its terms get smaller and smaller as \( n \) increases, eventually summing to a definite number. This transformation from a potentially infinite process to a finite answer is what makes convergence a powerful and essential concept in mathematical analysis.

Divergent Series

While convergence deals with series that settle into a single finite value, divergent series do not behave in this tame manner. A series is divergent if it does not have a finite limit. This means that as you add terms, the sum either keeps growing indefinitely, oscillates, or behaves erratically without reaching a well-defined value.

One common divergent series example is the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \). As you add more terms from this series, the sum keeps increasing without ever approaching a finite number. Divergent series are just as important as convergent ones, as they help shape our understanding of limits and boundaries within different mathematical structures. Understanding the nature of divergence helps students distinguish between what can be practically summed and what remains elusive.

Comparison Series

Comparison series are essential tools in the realm of infinite series, particularly when using tests like the Limit Comparison Test. The idea is to compare a series of interest with another series whose convergence or divergence we already understand.

When utilizing the Comparison Test, we select a known reference series. For example, with the series \( \sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3} \), we compare it to the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), a convergent p-series. The heart of the test involves computing the limit of the ratio of terms from each series as \( n \) approaches infinity. If the limit is a positive finite number, both series will converge or diverge together.

This comparison method is valuable because it allows mathematicians to conclude about a complex series by relating it to a simpler, well-understood series, smoothing the path to understanding its behavior.