Question

Determine the convergence of the given series. State the test used; more than one test may be appropriate. $$\sum_{n=1}^{\infty} \frac{n^{2}+4 n-2}{n^{3}+4 n^{2}-3 n+7}$$

Short answer

The series diverges by the Limit Comparison Test with the harmonic series.

Step-by-step solution

Identify the Type of Series

The given series is \( \sum_{n=1}^{\infty} \frac{n^{2}+4n-2}{n^{3}+4n^{2}-3n+7} \). This series has a rational function in the terms \( a_n = \frac{n^2 + 4n - 2}{n^3 + 4n^2 - 3n + 7} \). We will analyze the degrees of the polynomial in the numerator and denominator to choose a suitable test.

Determine the Dominant Terms

For large \( n \), the dominant term in the numerator is \( n^2 \) and in the denominator is \( n^3 \). As \( n \to \infty \), the term \( a_n \approx \frac{n^2}{n^3} = \frac{1}{n} \). This suggests that the series behaves like the harmonic series \( \sum \frac{1}{n} \), which is divergent.

Apply the Limit Comparison Test

Use the Limit Comparison Test with \( b_n = \frac{1}{n} \) (the harmonic series). Calculate the limit:\[ \lim_{{n \to \infty}} \frac{a_n}{b_n} = \lim_{{n \to \infty}} n \cdot \frac{n^2+4n-2}{n^3+4n^2-3n+7} = \lim_{{n \to \infty}} \frac{n^3 + 4n^2 - 2n}{n^3 + 4n^2 - 3n + 7} \]Simplifying gives:\[ \lim_{{n \to \infty}} \frac{1 + \frac{4}{n} - \frac{2}{n^2}}{1 + \frac{4}{n} - \frac{3}{n^2} + \frac{7}{n^3}} = 1 \]Since the limit is finite and non-zero, both series either converge or diverge together. Since \( \sum \frac{1}{n} \) diverges, \( \sum a_n \) also diverges.

Key concepts

Limit Comparison Test

The Limit Comparison Test is a useful tool in determining the convergence of series, particularly when dealing with rational functions. When you have two series, say \( \sum a_n \) and \( \sum b_n \), you can use this test as follows:

• Calculate \( \lim_{{n \to \infty}} \frac{a_n}{b_n} \).
• If the limit is a positive finite number, then both series will either converge or diverge together.

In our original problem, the given series \( \sum \frac{n^{2}+4n-2}{n^{3}+4n^{2}-3n+7} \) was compared to the harmonic series \( \sum \frac{1}{n} \). The comparison limit was found to be 1, indicating that since the harmonic series is known to diverge, our original series also diverges. This conclusion is a powerful result that can simplify the process of analyzing series, provided you pick a good comparison series.

Rational Functions

A rational function is any function that can be expressed as the ratio of two polynomials. In mathematics, rational functions are often dealt with because they can represent complex relationships easily and are highly applicable across calculus and algebra.

• The numerator is a polynomial of degree \( m \), while the denominator is a polynomial of degree \( n \).
• The behavior of these functions often depends heavily on their degree, meaning the highest powers of \( n \) in the numerator and denominator.

In the context of series like \( \sum \frac{n^{2}+4n-2}{n^{3}+4n^{2}-3n+7} \), determining the dominant terms in these polynomials helps us identify a simpler comparable series that shares similar convergence characteristics. Thus, rational functions are a foundational element in understanding series behavior as they often define the dominant form of complex mathematical sequences.

Divergent Series

Series can be categorized as either convergent or divergent, which describes their behavior as more terms are added:

• A convergent series approaches a specific value, meaning it has a finite sum.
• A divergent series, on the other hand, increases indefinitely and does not settle on any specific total.

The original problem focused on a series that diverges. The harmonic series, \( \sum \frac{1}{n} \), is a classic example of a divergent series. It does not approach any limit despite the addition of countless terms. The use of the Limit Comparison Test relies heavily on the understanding of divergent series, as you are often attempting to contextualize the series of interest within known behaviors like divergence or convergence.

Dominant Terms

Identifying dominant terms is crucial when analyzing series, particularly those formed by rational functions. The dominant term in a polynomial is the term with the highest power; it influences the series' behavior as \( n \) becomes very large:

• For example, in the polynomial \( n^2 + 4n - 2 \), the dominant term is \( n^2 \), while for \( n^3 + 4n^2 - 3n + 7 \), it's \( n^3 \).
• Dominant terms help simplify complex expressions, enabling us to find approximations like \( \frac{n^2}{n^3} = \frac{1}{n} \).

By focusing on the dominant terms, we can compare the series to simpler series, like the harmonic series, to gauge convergence or divergence more easily. This simplification process allows for more manageable calculations and can provide insights into the broader behavior of complex series expressions.