Problem 31 - Thomas Calculus

Question

Which of the series in Exercises \(15 - 48\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { n + 1 } $$

Step-by-Step Solution

Answer

The series diverges as it fails both the alternating series test and the absolute convergence test.

Step 1: Identify the Type of Series

The series given is \( \sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1} \). It is an alternating series because of the factor \((-1)^n\).

Step 2: Absolute Convergence Test

To check for absolute convergence, consider the absolute series \( \sum_{n=1}^{\infty} \left| (-1)^n \frac{n}{n+1} \right| = \sum_{n=1}^{\infty} \frac{n}{n+1} \). Since \( \frac{n}{n+1} \approx 1 \) as \( n \to \infty \), the series \( \sum_{n=1}^{\infty} \frac{n}{n+1} \) diverges as it behaves like the harmonic series.

Step 3: Use Alternating Series Test

Apply the Alternating Series Test. Consider \( a_n = \frac{n}{n+1} \). For the Alternating Series Test to apply, \( a_{n+1} \leq a_n \) and \( \lim_{n \to \infty} a_n = 0 \). Here, \( a_n \) does not converge to 0 since \( \lim_{n \to \infty} \frac{n}{n+1} = 1 eq 0 \).

Step 4: Conclude on Convergence

Since the absolute series diverges and the alternating series test fails because \( \lim_{n \to \infty} \frac{n}{n+1} eq 0 \), the original series \( \sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1} \) diverges.

Key Concepts

Alternating Series

An alternating series is a series whose terms alternate in sign, typically characterized by expressions like \((-1)^n\) or \((-1)^{n+1}\). This means that the series will have positive and negative terms in consecutive order.

To determine if an alternating series converges, we often apply the Alternating Series Test. According to this test, an alternating series \( \sum_{n=1}^{\infty} (-1)^n a_n \) converges if:

• \(a_n \) is decreasing, i.e., \(a_{n+1} \leq a_n\).
• The limit of \(a_n\) as \(n\) approaches infinity is zero, or \(\lim_{n \to \infty} a_n = 0\).

If these conditions are met, the series converges. However, if the limit \(a_n\) does not approach zero, as in the case of \(\frac{n}{n+1}\), the series does not satisfy the test and may diverge.

Absolute Convergence

Absolute convergence is a stronger form of convergence than regular convergence. A series \( \sum_{n=1}^{\infty} a_n \) is said to converge absolutely if the series of absolute values \( \sum_{n=1}^{\infty} |a_n| \) converges.

In other words, when you take away any alternating or negative signs and the series still converges, it is absolutely convergent.

• If a series converges absolutely, it means every rearrangement of the series also converges to the same sum.
• This is more robust than ordinary convergence where rearrangements can lead to different sums or even divergence.

For the specific problem given, the series \( \sum_{n=1}^{\infty} \left| (-1)^n \frac{n}{n+1} \right| = \sum_{n=1}^{\infty} \frac{n}{n+1} \) diverges. Therefore, it does not converge absolutely.

Divergence

A series diverges if it does not converge to a finite limit. There are many tests to assess divergence; if a series fails to meet the conditions of these tests for convergence, it diverges.

• The series \( \sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1} \) is a prime example. It fails the Alternating Series Test because the sequence \( a_n=\frac{n}{n+1} \) does not converge to zero.
• Additionally, its absolute series behaves like the harmonic series, which we know diverges.

It’s essential when analyzing a series to apply appropriate tests for both convergence and divergence, as failing one type of test can often provide clues about the series' behavior under other types of convergence tests.

Harmonic Series

The harmonic series is one of the most famous divergent series in mathematics. Written as \( \sum_{n=1}^{\infty} \frac{1}{n} \), this series grows without bound as more terms are added.

When we discuss other series in relation to the harmonic series, it is usually to illustrate a comparison test.

• A series that behaves like the harmonic series in its growth pattern, such as \( \sum_{n=1}^{\infty} \frac{n}{n+1} \), will also diverge.
• Understanding the harmonic series helps in predicting the behavior of similar series, especially when comparing growth rates or using limit comparison tests.

If a series behaves too similarly to the harmonic series, it can be a strong indication of divergence.