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Magazine Chapter 9: Problem 35
State whether the given series is absolutely convergent, conditionally convergent, or divergent. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{3^{n}}{2^{n+8}}$$
Short Answer
Expert verified
The series is divergent.
Step by step solution
01
Identify the Absolute Value of the Terms
Given the series \( \sum_{n=1}^{\infty}(-1)^{n} \frac{3^{n}}{2^{n+8}} \), we first consider the absolute value of the terms, which is \( \frac{3^{n}}{2^{n+8}} \). Removing the alternating sign \((-1)^n\) makes the series:\[ \sum_{n=1}^{\infty} \frac{3^{n}}{2^{n+8}} = \sum_{n=1}^{\infty} \frac{3^{n}}{2^n \cdot 2^8} = \frac{1}{2^8} \sum_{n=1}^{\infty} \left( \frac{3}{2} \right)^n. \]
02
Analyze the Geometric Series
The series \( \sum_{n=1}^{\infty} \left( \frac{3}{2} \right)^n \) is a geometric series with common ratio \( r = \frac{3}{2} \). For a geometric series \( \sum_{n=1}^{\infty} ar^n \), it converges if \( |r| < 1 \). Here, \( |\frac{3}{2}| > 1 \), so this series diverges.
03
Conclude on Absolute Convergence
Since the series \( \sum_{n=1}^{\infty} \left( \frac{3}{2} \right)^n \) diverges, the original series \( \sum_{n=1}^{\infty} |(-1)^n \frac{3^{n}}{2^{n+8}}| \) diverges. Therefore, the original series is not absolutely convergent.
04
Apply the Alternating Series Test
We apply the Alternating Series Test to the original series \( \sum_{n=1}^{\infty} (-1)^n \frac{3^{n}}{2^{n+8}} \). For this test, \( b_n = \frac{3^n}{2^{n+8}} \), which must satisfy: \( b_n \) is decreasing and \( \lim_{n \to \infty} b_n = 0 \). As \( n \to \infty \), \( \frac{3^n}{2^{n+8}} \to \frac{3}{4096} \to 0 \), but it does not guarantee convergence because \( b_n \) is not decreasing as required.
05
Conclusion on Conditional Convergence
Since \( b_n = \frac{3^n}{2^{n+8}} \) is not decreasing, the Alternating Series Test fails. Hence, the series is not conditionally convergent.
06
Final Conclusion on the Series
Based on the analysis above, the series \( \sum_{n=1}^{\infty} (-1)^n \frac{3^{n}}{2^{n+8}} \) is divergent.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergence Tests
Convergence tests are essential tools that help us determine whether a series converges or diverges. When given a series, checking if it converges (or diverges) is a crucial first step in understanding its behavior.Convergence tests come in various forms, each best suited for different types of series:
- Geometric Series Test: This test examines series of the form \( \sum ar^n \) and depends on the common ratio \( r \). A geometric series converges if the absolute value of \( r \) is less than 1. However, if \(|r| \geq 1\), the series diverges.
- Alternating Series Test: Used to decide if series with alternating terms converge. For this test to show convergence, the terms \( b_n \) must consistently decrease and approach zero as \( n \) goes to infinity.
Geometric Series
A geometric series is a series where each term is a constant multiple (called the common ratio) of the previous term. This kind of series holds great importance as it repeatedly occurs in various mathematical contexts.The formula for a geometric series is:\[ \sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + \cdots \]where \( a \) is the initial term and \( r \) is the common ratio. A critical feature of geometric series:
- The series converges if \( |r| < 1 \). The sum can be calculated using the formula \( S = \frac{a}{1-r} \).
- If \( |r| \geq 1 \), the series diverges. This means it doesn't settle to a fixed value as you sum more terms.
Alternating Series Test
The Alternating Series Test is a valuable tool for assessing series that alternate in sign. It provides a straightforward method to check for convergence under specific conditions.To apply the Alternating Series Test, a series in the form of \( \sum (-1)^n b_n \) needs to satisfy:
- The sequence \( b_n \) must be decreasing, meaning each term is less than or equal to the preceding term.
- The limit of \( b_n \) as \( n \) approaches infinity should be zero. This means as you look further along the series, the terms should ideally shrink towards zero.
