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When determining if a series converges absolutely, the Absolute Convergence Test is a good starting point. This test involves taking the absolute value of the series’ terms. For the series \( \sum_{n=3}^{\infty} \frac{(-1)^{n}}{10 \sqrt{n}} \), we consider the absolute series:

• \( \sum_{n=3}^{\infty} \left| \frac{(-1)^{n}}{10 \sqrt{n}} \right| = \sum_{n=3}^{\infty} \frac{1}{10 \sqrt{n}} \)

This simplifies to \( \frac{1}{10}\sum_{n=3}^{\infty} \frac{1}{\sqrt{n}} \). Here, the presence of \( \frac{1}{\sqrt{n}} \) indicates a p-series with \( p = \frac{1}{2} \), which leads us to explore whether this p-series converges. If \( p \leq 1 \), the p-series diverges, which means the original series does not converge absolutely. Understanding this allows us to progress by applying other convergence tests on the original series.

After finding that a series does not converge absolutely, the Alternating Series Test can help us determine if it converges conditionally. This test is applicable when a series has alternating positive and negative terms. Our series, \( \sum_{n=3}^{\infty} \frac{(-1)^{n}}{10 \sqrt{n}} \), is alternating because of the \((-1)^{n}\) factor.The Alternating Series Test has two key conditions:

• The terms \( a_n = \frac{1}{\sqrt{n}} \) must be decreasing.
• The limit of \( a_n \) as \( n \) approaches infinity must be zero.

In this case, \( \frac{1}{\sqrt{n}} \) is a decreasing sequence because \( \sqrt{n} \) increases as \( n \) grows. Additionally, \( \frac{1}{\sqrt{n}} \to 0 \) as \( n \to \infty \). Both conditions are satisfied, therefore the series converges conditionally.

The p-Series Test is crucial in analyzing series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \). The behavior of these series is determined by the exponent \( p \).- When \( p > 1 \), the p-series converges.- When \( p \leq 1 \), the p-series diverges.The series \( \sum_{n=3}^{\infty} \frac{1}{\sqrt{n}} \) can be written as \( \sum_{n=3}^{\infty} \frac{1}{n^{1/2}} \), which is a p-series with \( p = \frac{1}{2} \). Since \( p \leq 1 \), this tells us the series diverges. Recognizing this allows us to infer that the absolute series diverges, and combined with conditional convergence checks of other tests, it ultimately informs us about the nature of the original series' convergence.