91Ó°ÊÓ

Absolute convergence is a stronger form of convergence for series. In simple terms, a series \( \sum a_n \) converges absolutely if the series formed by taking the absolute values of its terms \( \sum |a_n| \) also converges. Why is this important? If a series converges absolutely, it guarantees convergence of the series itself, regardless of the signs of the terms. It's like having a solid foundation that supports the entire structure of a building.

In the given exercise, the series \( \sum_{n=1}^{\infty}(-1)^{n} \frac{\sin n}{n^{2}} \) is examined for absolute convergence. This involves analyzing \( \sum_{n=1}^{\infty} \left| (-1)^{n} \frac{\sin n}{n^{2}} \right| = \sum_{n=1}^{\infty} \frac{|\sin n|}{n^{2}} \). Since \(|\sin n| \leq 1\) for all \(n\), this gives us a comparison to the known convergent series \( \sum_{n=1}^{\infty} \frac{1}{n^{2}} \), which is a p-series with \(p=2\). With \(p>1\), we establish that the series converges absolutely. This means any rearrangement of terms will also converge, maintaining the sum finite and defined.

P-series are a particular kind of series that have the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \(p\) is a positive constant. These series are essential in understanding convergence behavior based on the value of \(p\). The convergence of a p-series depends solely on the parameter \(p\):

• Converges if \(p > 1\)
• Diverges if \(p \leq 1\)

In the context of the exercise, we see a similar structure in the absolute value terms: \( \sum_{n=1}^{\infty} \frac{|\sin n|}{n^{2}} \). Even though \(\sin n\) introduces variability in each term, its impact is bounded because \(|\sin n| \leq 1\). As a result, the effect on convergence follows that of a standard p-series with \(p=2\). Therefore, knowing the properties of p-series helps us quickly determine that our examined series behaves much like \( \sum_{n=1}^{\infty} \frac{1}{n^2} \), confirming it converges.

The Alternating Series Test provides a practical tool to determine the convergence of series that have terms changing sign, like the alternating series \( \sum (-1)^n a_n \). This test checks two simple conditions for convergence:

• The absolute value of the terms \(a_n\) must decrease steadily: \(a_{n+1} \leq a_n\).
• The limit of these terms as \(n\) tends towards infinity should be zero: \(\lim_{n \to \infty} a_n = 0\).

Applying this to the series \( \sum_{n=1}^{\infty}(-1)^{n} \frac{\sin n}{n^{2}} \), we analyze \(a_n = \frac{\sin n}{n^2}\). The function \(\sin n\) is bounded, and since \(n^2\) increases indefinitely, \(a_n\) clearly tends to 0. For the decreasing condition, since \((n+1)^2 > n^2\), we know that \(\frac{1}{(n+1)^2} \leq \frac{1}{n^2}\), ensuring this decrease. Thus, both conditions satisfy the alternating series test, confirming convergence.