Question

In the following exercises, use an appropriate test to determine whether the series converges. $$ \sum_{n=1}^{\infty} \frac{(n+1)}{n^{3}+n^{2}+n+1} $$

Short answer

The series converges by the Limit Comparison Test with \( \sum \frac{1}{n^2} \).

Step-by-step solution

Identify the Type of Series and Apply a Suitable Test

The series given is \( \sum_{n=1}^{\infty} \frac{n+1}{n^3+n^2+n+1} \). This resembles a rational function. In such cases, the Limit Comparison Test or the Direct Comparison Test is appropriate to determine convergence or divergence.

Simplify the Dominant Terms

Note that for large \( n \), the terms \( n^3 \) and \( n+1 \) are dominant. So, the fraction simplifies to \( \frac{n+1}{n^3} \approx \frac{n}{n^3} = \frac{1}{n^2} \). This suggests comparing it with the simpler series \( \sum \frac{1}{n^2} \).

Apply the Limit Comparison Test

Consider the two sequences: \( a_n = \frac{n+1}{n^3+n^2+n+1} \) and \( b_n = \frac{1}{n^2} \). Calculate the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{n+1}{n^3+n^2+n+1}}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{n^2(n+1)}{n^3+n^2+n+1} \). Simplifying further gives \( \lim_{n \to \infty} \frac{n^3+n^2}{n^3+n^2+n+1} \).

Evaluate the Limit

Divide each term by \( n^3 \) to get \( \lim_{n \to \infty} \frac{1+\frac{1}{n}}{1+\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}} \), which simplifies to \( \lim_{n \to \infty} \frac{1+\frac{1}{n}}{1} \equiv 1 \).

Determine the Convergence Using the Limit

Since \( \lim_{n \to \infty} \frac{a_n}{b_n} = 1 \) and \( \sum \frac{1}{n^2} \) is a convergent p-series (with \( p = 2 > 1 \)), by the Limit Comparison Test, \( \sum_{n=1}^{\infty} \frac{n+1}{n^3+n^2+n+1} \) also converges.

Key concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool for determining the convergence or divergence of an infinite series. It is particularly useful when dealing with series that include rational functions, like the one in our exercise. For this test, we compare a complex series to a simpler known series, usually a p-series, which is more straightforward to analyze.

Here's how the test works:

• Take the given series and identify or approximate the dominant terms that lead the behavior of the series as it approaches infinity.
• Select a comparison series. This step requires identifying a similar series whose convergence properties are already known.
• Compute the limit of the ratio of the series term to the comparison series term as n approaches infinity.

If the limit exists and is a finite number greater than zero, both series either converge or diverge together. In our specific exercise, we used the series \(\frac{1}{n^2}\) as our comparison, which is known to converge.
By comparing \(a_n = \frac{n+1}{n^3+n^2+n+1}\) and \(b_n = \frac{1}{n^2}\), we found their limit to be 1. Since the comparison series converges, our original series also converges by the Limit Comparison Test.

p-Series

A p-series is a type of series of the form \(\sum \frac{1}{n^p}\), where \(p\) is a real number. They are fundamental to understanding series convergence, as they offer a straightforward example for comparing more complicated series.

The convergence of a p-series depends on the value of \(p\):

• If \(p > 1\), the series converges. This means the terms are decreasing sufficiently fast as n increases, leading to a finite sum.
• If \(p \leq 1\), the series diverges. Here, the terms do not decrease fast enough, resulting in an infinite sum.

In the context of our exercise, we compared our given series with a p-series where \(p = 2\). Since 2 is greater than 1, the series \(\sum \frac{1}{n^2}\) converges. Knowing this property allowed us to confidently apply the Limit Comparison Test and conclude about the behavior of the original series.

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. In series analyses, especially those involving convergence, recognizing a rational function's dominant terms becomes crucial.

For large values of \(n\), higher-power terms in the polynomials tend to dominate the behavior of the function. This means:

• For a rational function like \(\frac{P(n)}{Q(n)}\), where \(P(n)\) and \(Q(n)\) are polynomials, the polynomial with the highest degree in the numerator and denominator will largely dictate the function's behavior as n becomes very large.
• Simplifying these terms helps us approximate the function with a more familiar series, like a p-series.

In our exercise, we identified \(\frac{n+1}{n^3 + n^2 + n + 1}\) as a rational function. By recognizing \(n^3\) and \(n+1\) as the leading terms as \(n\) increases, we simplified the analysis greatly. This simplification allowed us to use the Limit Comparison Test effectively, using the simple convergence properties of a p-series.