91Ó°ÊÓ

Absolute convergence means that if we take the absolute value of each term in a series and the resulting series converges, then the original series also converges. It is a stronger form of convergence which applies to both alternating and non-alternating series.
To determine absolute convergence, we examine the series of absolute values. For the given series, it's \( \sum_{n=1}^{\infty} \left| (-1)^{n+1} \frac{n}{n+3} \right| = \sum_{n=1}^{\infty} \frac{n}{n+3} \).
This resembles the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), known to diverge. Since this absolute value series diverges, the original series does not converge absolutely.
In essence, if a series does not converge absolutely, it might still converge conditionally. However, this is not the case here as we’ll explore in the following sections.

Conditional convergence occurs when a series converges, but does not converge absolutely. This typically happens in alternating series, where even though the absolute series diverges, the original series might still converge due to the alternating nature of its terms.
However, for a series to be conditionally convergent, it must pass the alternating series test which we'll detail later. Our series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n+3} \) fails this test. This means it neither converges absolutely nor conditionally.
Thus, for our series, neither condition holds, so we conclude it diverges completely.

The Alternating Series Test is a handy tool for checking the convergence of series with alternating signs. For a given series \( \sum (-1)^n a_n \), this test stipulates two conditions for convergence:

• Each subsequent term \( a_n \) must be smaller than the previous term (i.e., \( a_n \) must be decreasing).
• The limit of \( a_n \) as \( n \) approaches infinity must be zero.

Our series is \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n+3} \) with \( a_n = \frac{n}{n+3} \). Although it alternates in sign, as \( n \to \infty \), \( a_n \to 1 \), not zero, so it does not satisfy the necessary conditions.
Since the alternating series test fails, the series does not converge via this method, further supporting our findings on divergence.

Divergence simply means that the sum of an infinite series does not approach a finite limit. In the case of the series \( \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n+3} \), we found that it neither converges absolutely nor conditionally.
Because both the alternating series test and the series of absolute values indicate divergence, we conclude that the overall series diverges.
Divergence should not be viewed negatively but rather as a result of specific criteria not being met. For this series, the failure arises because its terms do not approach zero and the absolute series diverges.