Problem 10
Question
In Exercises \(1 - 14 ,\) determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { \ln n } $$
Step-by-Step Solution
Verified Answer
The series converges by the Alternating Series Test.
1Step 1: Identify the Series Pattern
The series given is \( \sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { \ln n } \). This is an alternating series because it involves \( ( - 1 ) ^ { n + 1 } \), implying that the terms alternate in sign.
2Step 2: Check Alternating Series Test Conditions - Non-Increasing Sequence
To apply the Alternating Series Test, we confirm if \( \frac{1}{\ln n} \) is a non-increasing sequence (i.e., \( a_{n+1} \leq a_{n} \)). Since \( \ln(n) \) grows as \( n \) increases, \( \frac{1}{\ln n} \) decreases. Hence, \( \frac{1}{\ln n} \) is a non-increasing sequence for \( n \geq 2 \).
3Step 3: Check Alternating Series Test Conditions - Limit Condition
Next, we check if \( \lim_{n \to \infty} \frac{1}{\ln n} = 0 \). As \( n \to \infty \), \( \ln n \to \infty \), making \( \frac{1}{\ln n} \to 0 \). This satisfies the limit condition of the Alternating Series Test.
4Step 4: Conclusion of Alternating Series Test
Since both conditions are met (non-increasing and limit condition), the Alternating Series Test confirms that the series \( \sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 1 } { \ln n } \) converges.
Key Concepts
Convergence of SeriesNon-Increasing SequenceLimit of a SequenceLogarithmic Functions
Convergence of Series
A series is a sum of terms in a sequence. The convergence of a series determines whether this sum approaches a finite value as more terms are added. When dealing with an alternating series, such as the one in the given exercise, the terms sequentially change their signs.
Convergence occurs if, as you add more terms, the sum gets closer to a specific number. There are different tests to determine convergence, and the Alternating Series Test is one such tool. This test applies specifically to series where the terms alternate in sign and helps us decide if the series converges.
For an alternating series to converge, two main criteria must be met:
Convergence occurs if, as you add more terms, the sum gets closer to a specific number. There are different tests to determine convergence, and the Alternating Series Test is one such tool. This test applies specifically to series where the terms alternate in sign and helps us decide if the series converges.
For an alternating series to converge, two main criteria must be met:
- The sequence of absolute values of the terms must be non-increasing.
- The limit of these absolute values must approach zero.
Non-Increasing Sequence
A non-increasing sequence is one where each term is not greater than the previous term. In mathematical terms, a sequence \( a_n \) is non-increasing if \( a_{n+1} \leq a_n \) for all \( n \).
For our series, we have a sequence \( \frac{1}{\ln n} \). To check if it's non-increasing, we need to look at its behavior as \( n \) increases. Since the natural logarithm \( \ln n \) increases with \( n \), the reciprocal \( \frac{1}{\ln n} \) decreases. Therefore, the sequence is non-increasing for \( n \geq 2 \).
Non-increasing sequences are important in the study of series because they help us apply convergence tests like the Alternating Series Test. Ensuring a sequence is non-increasing is often the first step in evaluating its convergence properties.
For our series, we have a sequence \( \frac{1}{\ln n} \). To check if it's non-increasing, we need to look at its behavior as \( n \) increases. Since the natural logarithm \( \ln n \) increases with \( n \), the reciprocal \( \frac{1}{\ln n} \) decreases. Therefore, the sequence is non-increasing for \( n \geq 2 \).
Non-increasing sequences are important in the study of series because they help us apply convergence tests like the Alternating Series Test. Ensuring a sequence is non-increasing is often the first step in evaluating its convergence properties.
Limit of a Sequence
The concept of a limit is fundamental in calculus and analysis, describing the behavior of a sequence as \( n \) approaches infinity. For a sequence \( a_n \), the limit \( \lim_{n \to \infty} a_n = L \) means that as \( n \) becomes very large, the sequence approaches a number \( L \).
In our exercise, we look at the limit of \( \frac{1}{\ln n} \) as \( n \to \infty \). As \( n \) increases, \( \ln n \) grows without bound, making \( \frac{1}{\ln n} \) approach zero. This indicates that the terms of the series become very small as \( n \) gets larger.
Checking the limit of a sequence is crucial in determining series convergence, especially with the Alternating Series Test, where one condition is that the terms of the sequence must approach zero.
In our exercise, we look at the limit of \( \frac{1}{\ln n} \) as \( n \to \infty \). As \( n \) increases, \( \ln n \) grows without bound, making \( \frac{1}{\ln n} \) approach zero. This indicates that the terms of the series become very small as \( n \) gets larger.
Checking the limit of a sequence is crucial in determining series convergence, especially with the Alternating Series Test, where one condition is that the terms of the sequence must approach zero.
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions and have a wide range of applications in mathematics. In our problem, the logarithmic function \( \ln n \) plays a key role in the behavior of the sequence \( \frac{1}{\ln n} \).
The natural logarithm \( \ln n \) increases as \( n \) increases, but it does so very slowly compared to exponential growth. This slow growth rate means that for large \( n \), \( \ln n \) becomes significant but not overwhelmingly large in relation to \( n \) itself.
Understanding logarithmic functions is important because they often appear in calculus, especially in problems dealing with growth and decay. In series problems, recognizing how the logarithmic function affects the terms of a sequence can assist us in applying convergence tests and understanding the overall behavior of the series.
The natural logarithm \( \ln n \) increases as \( n \) increases, but it does so very slowly compared to exponential growth. This slow growth rate means that for large \( n \), \( \ln n \) becomes significant but not overwhelmingly large in relation to \( n \) itself.
Understanding logarithmic functions is important because they often appear in calculus, especially in problems dealing with growth and decay. In series problems, recognizing how the logarithmic function affects the terms of a sequence can assist us in applying convergence tests and understanding the overall behavior of the series.
Other exercises in this chapter
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