Question

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

Short answer

The series converges by the Alternating Series Test.

Step-by-step solution

Identify the Series Type

We start by identifying the series as an alternating series due to the term \((-1)^{n+1}\). This indicates the series is alternating in sign.

Alternate Series Test Conditions

For an alternating series of the form \(\sum (-1)^n a_n\), to converge, \(a_n\) must satisfy two conditions: 1) \(a_{n+1} \leq a_n\) for all \(n\), and 2) \(\lim_{n \to \infty} a_n = 0\). Here, \(a_n = \frac{n+1}{n^3 + 3n^2 + 3n + 1}\).

Check Decreasing Sequence

To verify if \(a_n\) is decreasing, examine the expression \(\frac{n+1}{n^3 + 3n^2 + 3n + 1}\). As \(n\) increases, the denominator grows faster than the numerator, suggesting \(a_n\) decreases. Further verification involves checking \(a_{n+1} \leq a_n\).

Check the Limit as n Approaches Infinity

Calculate \(\lim_{n \to \infty} \frac{n+1}{n^3 + 3n^2 + 3n + 1} = 0\). As \(n\) approaches infinity, the highest degree term in the denominator dominates, thus making the limit of the fraction zero.

Conclusion Based on Tests

Since \(a_n\) is decreasing and \(\lim_{n \to \infty} a_n = 0\), both conditions of the alternating series test are satisfied. Thus, the series converges.

Key concepts

Convergence Tests

In mathematics, convergence tests play a crucial role in determining whether an infinite series converges or diverges. A series converges if the sum of its infinite terms approaches a finite number. One essential test for convergence is the **Alternating Series Test**. This test is specifically used for series that alternate in sign.

For an alternating series of the form \[\sum_{n=1}^{\infty} (-1)^{n+1}a_n,\]the series converges if two conditions are met:

• The sequence \(a_n\) is decreasing. This means that each term is smaller than the preceding term, mathematically shown as \(a_{n+1} \leq a_n\).
• The limit of \(a_n\) as \(n\) approaches infinity must be zero, represented as \(\lim_{n \to \infty} a_n = 0\).

In the exercise provided, the series passed the Alternating Series Test, meaning it converges to a certain value.

Infinite Series

An infinite series is a sum of an ordered sequence of terms continuously added without limit. It is written in the form\[\sum_{n=1}^{\infty} a_n,\]where \(a_n\) represents the general term of the series. Understanding infinite series is key in advanced mathematics and its applications, such as calculus and analysis.

Two major outcomes of studying an infinite series are determining if it converges to a finite number or diverges, which means the sum does not settle on a specific value. Recognizing the type of series, such as geometric, harmonic, or alternating, helps in choosing an appropriate method to analyze convergence.

In the provided exercise, an alternating series is at the center. Knowing this informs the choice of using the Alternating Series Test for convergence analysis.

Sequence Analysis

Sequence analysis involves studying the properties and behaviors of ordered lists of numbers, or sequences. This includes observing whether a sequence eventually becomes smaller, larger, or stays the same, and whether it approaches a limit as the term number increases.

In the context of the series from the exercise, analyzing the sequence \(a_n = \frac{n+1}{n^3 + 3n^2 + 3n + 1}\) is vital. This sequence must meet specific criteria for the series to converge according to the Alternating Series Test.

Firstly, the analysis involves verifying that \(a_n\) is decreasing. This is assessed by comparing \(a_n\) with \(a_{n+1}\) and observing the growth rates of the numerator and denominator. Secondly, the limit of \(a_n\) as \(n\) approaches infinity is checked. In this case, due to the dominating cubic term in the denominator, the limit is zero, indicating that the sequence behavior supports convergence.