91Ó°ÊÓ

The Alternating Series Test is a handy tool when deciding if a series is convergent. It applies to series where the terms change sign, meaning they alternate. An alternating series typically has the form

• \(-1^n a_n\)
• \(-1^{n+1} a_n\)

To conclude if such a series converges, there are two main conditions to check:
- The sequence \(a_n\) must approach zero as \(n\) approaches infinity.
- Each term \(a_n\) should be positive.
These simple checks ensure the oscillation in positive and negative values eventually leads to a sum creeping towards a stable number. When we apply this test to the series given \[\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n \sqrt{n}}\]we observe the terms \(\frac{1}{n^{3/2}}\). These terms are positive and become smaller and smaller until they nearly vanish as \(n\) increases. So, the series is conditionally convergent due to the alternating signs and decreasing terms.

Absolute convergence takes the notion of convergence one step further. A series \(\sum a_n\) is said to converge absolutely if the series of the absolute values \[\sum |a_n|\]also converges. This is a more powerful form of convergence.
Why is this important? If a series converges absolutely, it also converges in the regular sense. However, not all convergent series converge absolutely.
For our series, we consider the absolute series:\[\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}\]When you remove the signs, the result is a p-series, which we'll discuss next.
If this series converges, the original series converges absolutely.
In this case, we find it indeed converges due to the p-series behavior (as shown in the steps), confirming absolute convergence.

A p-series is an insightful concept when analyzing series of the form\[\sum_{n=1}^{\infty} \frac{1}{n^p}\]Here, the power \(p\) decides whether the series converges or diverges.

A p-series converges only if \(p > 1\). Conversely, if \(p \leq 1\), the series diverges.

In the case of our original exercise, we deal with \[\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}\]This series is a classic p-series with \(p = 3/2\), which is indeed greater than 1. Therefore, according to the p-series test, this series converges.
Using this insight, we conclude that the series given in the exercise converges absolutely, bolstering our understanding of absolute convergence previously discussed.