91Ó°ÊÓ

When we talk about absolute convergence, we aim to check whether the series would converge if we were to ignore the signs of the terms. Essentially, we use the absolute value of each term in the series and see if the resulting series converges. For example, consider the series \[\sum (-1)^{n+1} \frac{n}{n+3}\] To assess absolute convergence, we look at the series: \[\sum \left| (-1)^{n+1} \frac{n}{n+3} \right| = \sum \frac{n}{n+3}\]If this series converges, so does the original one. However, the term \( \frac{n}{n+3} \) does not tend to zero as \( n \to \infty \); in fact, it approaches 1. Hence, the series does not converge absolutely. Checking for absolute convergence first helps simplify things, letting us know whether the series might be tackled with straightforward, non-sign-based methods.

Conditional convergence occurs if a series converges, but does not converge absolutely. After ruling out absolute convergence, we might still want to explore whether the series converges conditionally. Unlike absolute convergence, conditional convergence considers the actual nature of the terms—positive or negative.Following our example series:\[\sum (-1)^{n+1} \frac{n}{n+3}\]We've already determined it doesn't converge absolutely. If the general term \( a_n \) tends to zero and alternates in sign appropriately, the series might converge conditionally. However, for this to happen:

• The terms must approach zero \( \lim_{n\to\infty}a_n=0 \).
• The terms must be non-increasing \( a_{n+1} \leq a_n \).

In this case, because \( \frac{n}{n+3} \) approaches 1, it does not satisfy these conditions, and therefore, the series does not converge, neither conditionally nor absolutely.

The Alternating Series Test is a useful tool to determine if a series converges based upon the nature of its terms. Specifically, it applies to series where the terms alternate in sign, such as our example series.For the test to assert convergence, the sequence of the terms must satisfy two conditions:

• The sequence \( a_n \) is non-increasing.
• The limit of the sequence \( a_n \) as \( n \) tends to infinity is zero \( \lim_{n\to\infty}a_n=0 \).

Examining our series \[\sum (-1)^{n+1} \frac{n}{n+3}\]We recognize \( a_n = \frac{n}{n+3} \). As \( n \) endless grows, this term moves toward 1, failing the second condition of the test.Thus, the series is not convergent by the Alternating Series Test. This inability to converge proves that our series doesn't have conditional convergence either.