Problem 10 - Calculus: One and Several Variables

Question

Determine whether the series converges or diverse. $$\sum \frac{\ln k}{k^{3}}$$

Step-by-Step Solution

Answer

The series \( \sum \frac {\ln k} {k^{3}} \) converges.

Step 1: Simplify the Series

Notice that the series \( \sum \frac {\ln k} {k^{3}} \) has elements that when \( k > e \), can be approximated as less than \( \frac {1} {k^{2}} \). Meaning that \( \frac {\ln k} {k^{3}} < \frac {1} {k^{2}} \) for \( k > e \).

Step 2: Comparison Test

Use the comparison test. The series \( \sum \frac {1} {k^{2}} \) is a p-series with \( p = 2 \) and it is known that a p-series with \( p > 1 \) converges.

Step 3: Conclusion

Thus, by the comparison test, if \( 0 \leq a_{k} \leq b_{k} \) for all \( k \geq N \) and \( \sum b_{k} \) converges, then \( \sum a_{k} \) also converges. Hence the series \( \sum \frac {\ln k} {k^{3}} \) converges.

Key Concepts

Comparison Test

The Comparison Test is a powerful tool in determining the convergence or divergence of series. It is particularly useful when you have a complicated series and want to compare it to a simpler, more familiar series. The idea is to take two series, say \( \sum a_{k} \) and \( \sum b_{k} \), and compare their terms.

• If \( 0 \leq a_{k} \leq b_{k} \) for all \( k \) from some point onward (say \( k \geq N \)), and \( \sum b_{k} \) converges, then \( \sum a_{k} \) also converges.
• Conversely, if \( a_{k} \geq b_{k} \) for all \( k \geq N \), and \( \sum b_{k} \) diverges, then \( \sum a_{k} \) also diverges.

This test is particularly handy when you can bound the given series by another series whose convergence is already known. It's like using a checkpoint; if this well-known series converges or diverges, and our series mimics it closely enough, our series will behave the same.

P-Series

P-Series are a special kind of series that have the form \( \sum \frac{1}{k^{p}} \) where \( p \) is a positive constant. These series are very useful as they provide a simple benchmark for determining convergence.The behavior of a p-series depends on the value of \( p \):

• If \( p > 1 \), the series converges.
• If \( p \leq 1 \), the series diverges.

The series \( \sum \frac{1}{k^{2}} \) serves as a classic example with \( p = 2 \). This series is known to converge. In our original problem, we used this known convergent p-series to compare and determine the convergence of another series.

Natural Logarithm

The Natural Logarithm function, often written as \( \ln x \), is the logarithm to the base \( e \) where \( e \) is an irrational constant approximately equal to 2.71828. This function is prevalent in mathematics, as it arises in various growth and decay problems, exponential changes, and calculus.In the context of series, \( \ln k \) grows slowly compared to polynomial terms like \( k^n \) where \( n > 0 \). This slow growth plays a significant role in evaluating the convergence or divergence of series that include logarithmic terms in their numerator, as such terms might be overshadowed by larger polynomial denominators.

Convergence of Series

Determining whether a series converges is a fundamental question in calculus and analysis. When a series converges, it means the sum of its infinite terms approaches a finite number. This is an essential concept since it helps understand the behavior of functions and datasets over intervals. There are several methods to test for convergence:

• Comparison Test: Comparing a given series to a known benchmark to draw conclusions.
• Ratio Test: Analyzing the ratio of successive terms for convergence.
• Integral Test: Utilizing improper integrals to assess convergence.

Exploring different techniques allows mathematicians and students to develop a robust understanding of series behavior. Convergence means reliability and predictability in calculations, crucial for consistent results in mathematics and applied fields.