Problem 48 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges absolutely, converges conditionally, or diverges. Explain your reasoning carefully. \(\sum_{n=3}^{\infty} \frac{(-1)^{n}}{10 \sqrt{n}}\)

Step-by-Step Solution

Answer

The given series converges conditionally but not absolutely as it passes the Alternating Series Test but fails the Absolute Convergence Test.

Step 1: Applying the Absolute Convergence Test

Calculate the absolute series \( \sum_{n=3}^{\infty} \left| \frac{(-1)^{n}}{10 \sqrt{n}} \right| = \sum_{n=3}^{\infty} \frac{1}{10 \sqrt{n}} = \frac{1}{10}\sum_{n=3}^{\infty} \frac{1}{\sqrt{n}} \). This is a p-series, with \( p = \frac{1}{2} \). Since \( p \leq 1 \), this series diverges by the P-Series Test. Thus, the original series doesn't converge absolutely.

Step 2: Applying the Alternating Series Test

Since the Absolute Convergence Test revealed the original series isn't absolutely convergent, apply the Alternating Series Test to determine if it's conditionally convergent. A series \( \sum_{n=3}^{\infty} (-1)^{n}a_{n} \) will converge if the following two conditions are met: 1) The terms \( a_{n} \) are decreasing, and 2) They approach to 0 as \( n \) goes to infinity. As \( \sqrt{n} \) is increasing with \( n \), \( \frac{1}{\sqrt{n}} \) is decreasing. It's obvious that \( \frac{1}{\sqrt{n}} \rightarrow 0 \) as \( n \rightarrow \infty \). Hence, both conditions are satisfied.

Key Concepts

Absolute Convergence Test

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.

Alternating Series Test

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.

p-Series Test

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.