Problem 20 - Calculus: Early Transcendentals

Question

Test the series for convergence or divergence. \( \displaystyle \sum_{k = 1}^{\infty} \frac {\sqrt [3]{k} - 1}{k (\sqrt{k} + 1)} \)

Step-by-Step Solution

Answer

The series converges by the p-series test and confirmed using the Limit Comparison Test.

Step 1: Simplify the Terms

The given series is \( \sum_{k = 1}^{\infty} \frac{\sqrt[3]{k} - 1}{k(\sqrt{k} + 1)} \). To analyze convergence, first consider simplifying the expression \( \frac{\sqrt[3]{k} - 1}{k(\sqrt{k} + 1)} \) for large \( k \). Since both terms in the numerator and denominator depend on \( k \), let's try simplifying it. The numerator approximately behaves like \( k^{1/3} \) and the denominator behaves like \( k^{3/2} \), giving us approximately \( \frac{k^{1/3}}{k^{3/2}} = k^{-7/6} \).

Step 2: Use the P-Series Test

The expression \(\frac{k^{1/3}}{k^{3/2}} = k^{-7/6}\) resembles a p-series form \( \sum \frac{1}{k^{p}} \) where \( p = \frac{7}{6} \). In the p-series test, a series \( \sum \frac{1}{k^p} \) converges if \( p > 1 \). In this case, since \( \frac{7}{6} > 1 \), the series converges.

Step 3: Verify using Limit Comparison Test

To confirm, use the Limit Comparison Test with \( b_k = \frac{1}{k^{7/6}} \). Compute the limit: \( \lim_{k\to\infty} \frac{\sqrt[3]{k} - 1}{k(\sqrt{k} + 1)} \bigg/ \frac{1}{k^{7/6}} \). This simplifies to \( \lim_{k\to\infty} \frac{(\sqrt[3]{k} - 1)k^{7/6}}{k^{3/2}(\sqrt{k} + 1)} = \lim_{k\to\infty} \frac{k^{1/3}}{k^{1/3}} = 1 \). Since the limit is finite and non-zero, by the Limit Comparison Test, both the original series and \( \sum \frac{1}{k^{7/6}} \) share the same convergence behavior. Therefore, the original series converges.

Key Concepts

P-Series Test

When dealing with infinite series, determining convergence is crucial. One fundamental tool for this purpose is the P-Series Test. This test involves series that have the form \[\sum \frac{1}{k^p},\]where \(p\) is a constant. Here is how the P-Series Test can be applied:

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

A series like \(\sum \frac{1}{k^{7/6}}\) is a classic example of a p-series. In this case, since \(p = \frac{7}{6} > 1\), the series will converge. The P-Series Test is a straightforward and powerful way to quickly ascertain the behavior of certain types of series. By converting a complex series into a p-series form, the convergence properties can be readily determined.

Limit Comparison Test

Sometimes, simplifying a series to exactly match a P-Series form might not be possible. That's where the Limit Comparison Test comes in. This test is useful when comparing the series in question with a known benchmark series that behaves similarly. Here's how it works:

• Choose a reference series \(b_k\) that is already known to converge or diverge.
• Calculate the limit \(\lim_{k\to\infty} \frac{a_k}{b_k}\), where \(a_k\) are the terms from the series you are analyzing.
• If the limit is a positive finite number, both series will have the same behavior regarding convergence or divergence.

For instance, the original series terms \(a_k = \frac{\sqrt[3]{k} - 1}{k(\sqrt{k} + 1)}\) are compared with \(b_k = \frac{1}{k^{7/6}}\). Calculating \[\lim_{k\to\infty} \frac{\frac{\sqrt[3]{k} - 1}{k(\sqrt{k} + 1)}}{\frac{1}{k^{7/6}}} = 1,\]we find this limit confirms the convergence of both series. The Limit Comparison Test is invaluable for cases where direct simplification isn't possible.

Series Simplification

Before applying any test for convergence or divergence, simplifying the series can be an effective strategy. Simplification can reveal hidden patterns or similarities to familiar forms, such as p-series. In the original problem, the series:\[\sum_{k = 1}^{\infty} \frac{\sqrt[3]{k} - 1}{k(\sqrt{k} + 1)},\]looks complicated at first glance. However, by focusing on the behavior of terms for large \(k\), the expression simplifies to\[\frac{k^{1/3}}{k^{3/2}} = k^{-7/6}.\]This simplification is crucial because it helps in identifying the series form, making it easier to use convergence tests like the P-Series Test or Limit Comparison Test. Understanding how to break down complex series terms is key to determining the convergence properties efficiently.

Mathematical Analysis

Mathematical analysis provides the underpinnings for rigorously studying series and their convergence. When analyzing series, one must deeply understand limit behavior, function approximation, and simplification techniques.

• The essence is to identify how terms behave as they approach infinity, which aids in determining overall series behavior.
• Tools like the P-Series Test and Limit Comparison Test, grounded in mathematical principles, allow investigation into whether infinite series converge or not.

For the given series, mathematical analysis starts with investigating the function's behavior, simplifies the algebra, and strategically compares terms using various tests. A sound analytical approach ensures that each step adheres to mathematical rigorations, providing robust conclusions about series convergence or divergence. Utilizing such analytical methods allows students to approach series exercises with confidence and clarity.