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Magazine Chapter 10: Problem 27
Use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}}$$
Short Answer
Expert verified
The series \( \sum_{n=1}^{\infty} \frac{\ln n}{n^3} \) converges by the Limit Comparison Test with a \( p\)-series.
Step by step solution
01
Identify the Series Type
We are given the series \( \sum_{n=1}^{\infty} \frac{\ln n}{n^{3}} \). This is an example of an infinite series where each term is in the form \( a_n = \frac{\ln n}{n^3} \). The series terms contain a logarithmic numerator and a polynomial denominator.
02
Select a Convergence Test
To analyze the convergence of series where terms contain logarithmic and polynomial elements, the Limit Comparison Test offers a helpful approach. We'll compare the given series with a simpler series whose convergence is already known, such as a \( p\)-series. Generally, \( \sum_{n=1}^{\infty} \frac{1}{n^{p}} \) converges if \( p > 1 \).
03
Choose a Comparable Series
Consider the series \( \sum_{n=1}^{\infty} \frac{1}{n^{2.5}} \). This series converges because it is a \( p\)-series with \( p = 2.5 > 1 \). This simpler series has a form similar to the given series, making it a good candidate for comparison.
04
Apply the Limit Comparison Test
The Limit Comparison Test relies on the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \), where \( a_n = \frac{\ln n}{n^3} \) and \( b_n = \frac{1}{n^{2.5}} \). Evaluate:\[\lim_{n \to \infty} \frac{\frac{\ln n}{n^3}}{\frac{1}{n^{2.5}}} = \lim_{n \to \infty} \frac{\ln n}{n^{3 - 2.5}} = \lim_{n \to \infty} \frac{\ln n}{n^{0.5}}.\]
05
Evaluate the Limit
To solve \( \lim_{n \to \infty} \frac{\ln n}{n^{0.5}} \), consider that as \( n \to \infty \), the \( n^{0.5} \) in the denominator grows much faster than \( \ln n \). Consequently, the limit simplifies to 0:\[\lim_{n \to \infty} \frac{\ln n}{n^{0.5}} = 0.\]
06
Conclusion on Convergence
Since \( \lim_{n \to \infty} \frac{\ln n}{n^{0.5}} = 0 \) and the series \( \sum_{n=1}^{\infty} \frac{1}{n^{2.5}} \) converges, by the Limit Comparison Test, the original series \( \sum_{n=1}^{\infty} \frac{\ln n}{n^3} \) also converges.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit Comparison Test
The Limit Comparison Test is a powerful tool when tackling the convergence of complex series. It allows us to draw parallels between a given series and another, simpler series whose behavior is already well-understood. Here鈥檚 the basic idea:
- Identify the series you want to test. Let's call this series \( \sum a_n \).
- Choose a simpler series \( \sum b_n \) that resembles \( \sum a_n \) in terms of growth. Usually, \( b_n \) is a well-known convergent or divergent series, like a \( p \)-series.
- Evaluate the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \). If this limit is a finite number greater than 0, both series \( \sum a_n \) and \( \sum b_n \) will either converge or diverge together.
Convergence Tests
Convergence tests are strategies employed to determine whether a given infinite series converges or diverges. There are several types of tests, each with its own conditions and scope of use:
- The Ratio Test: This examines the ratio of successive terms in the series and is effective for series underlying exponential terms.
- The Root Test: Similar to the Ratio Test but it assesses the \( n^{\text{th}} \) root of the terms, helpful for series with terms raised to powers.
- The Direct Comparison Test: Directly compares a series with another series with known convergence or divergence behavior.
- The Limit Comparison Test: As described earlier, it involves taking the limit of the ratio of terms from two series.
- The Integral Test: Useful for series represented by integrable functions, linking the convergence of an infinite series to an improper integral.
Infinite Series
An infinite series is a summation of an infinite sequence of terms. Infinite series can converge to a specific number, diverge, or become arbitrarily large or small. A deeper understanding of infinite series involves:
- **Convergence**: This occurs when the sum of the series approaches a finite limit. An example is the geometric series where the ratio of successive terms is less than 1.
- **Divergence**: When the series grows without bound or oscillates, it is deemed divergent. An example is the harmonic series \( \sum \frac{1}{n} \), which diverges despite diminishing terms.
- **Convergence Tests**: Applying these can determine the behavior of series. Tests like the Limit Comparison Test, Ratio Test, and others are essential tools.
