Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\sum_{k=1}^{\infty} \frac{3}{(4+3 k)^{7 / 6}}\)

Step-by-Step Solution

Answer

The series converges.

Step 1: Identify the Series Type

The given series is \( \sum_{k=1}^{\infty} \frac{3}{(4+3k)^{7 / 6}} \). Notice that this is an infinite series, where the general term is \( a_k = \frac{3}{(4+3k)^{7/6}} \). Given the structure \( a_k = \frac{C}{(ak+b)^p} \), this type of series is suitable for the Convergence Tests.

Step 2: Apply the Limit Comparison Test

For convergence of series of this form, we can use the Limit Comparison Test. We compare our series with the simpler series \( \sum_{k=1}^{\infty} \frac{1}{(3k)^{7/6}} \). The general term is \( b_k = \frac{1}{(3k)^{7/6}} \).

Step 3: Setup Limit for Comparison

Compute the limit \( \lim_{k \to \infty} \frac{\frac{3}{(4+3k)^{7/6}}}{\frac{1}{(3k)^{7/6}}} \). This simplifies to the computation of \( \lim_{k \to \infty} 3 \cdot \frac{(3k)^{7/6}}{(4+3k)^{7/6}} \).

Step 4: Simplify the Limit Expression

Factor out the highest power of \( k \) in both numerator and denominator: \( \lim_{k \to \infty} 3 \cdot \left(\frac{3^{7/6}k^{7/6}}{(3k)^{7/6}} \right) = 3 \cdot \left(\frac{3^{7/6}}{(3)^{7/6}} \right) \cdot \left(\frac{k^{7/6}}{k^{7/6}} \right) \cdot \left(\frac{1}{\left(1+\frac{4}{3k}\right)^{7/6}}\right) \).

Step 5: Evaluate the Limit

As \( k \to \infty \), \( \frac{4}{3k} \to 0 \). Thus, \( \left(1+\frac{4}{3k}\right)^{7/6} \to 1 \). Therefore, the limit becomes \( 3 \times 1 = 3 \).

Step 6: Determine Convergence or Divergence

Since the original series' terms behave like \( \sum \frac{1}{k^{7/6}} \) as \( k \to \infty \), and since \( 7/6 > 1 \), the series \( \sum \frac{1}{k^{7/6}} \) converges by the p-series test. Hence, the given series \( \sum \frac{3}{(4+3k)^{7/6}} \) also converges by the Limit Comparison Test.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool used when we want to determine the convergence or divergence of an infinite series. Picture yourself in a situation where you have a complicated series and you want to know its behavior. The Limit Comparison Test allows you to compare this series with another one that is simpler and easier to analyze.

Here's how it works: take two series, \( \sum a_k \) and \( \sum b_k \). You calculate the limit \( \lim_{k \to \infty} \frac{a_k}{b_k} \). If this limit exists and is a positive finite number (\( L eq 0 \) and \( L eq \infty \)), then both series will either converge or diverge together.

• If \( \sum b_k \) is known to converge, then \( \sum a_k \) will also converge.
• Conversely, if \( \sum b_k \) diverges, then \( \sum a_k \) will then also diverge.

In our example, by simplifying the comparison series to \( \sum \frac{1}{(3k)^{7/6}} \) and using the Limit Comparison Test, we found that our original series converges because it behaves similarly to the comparison series.

p-series test

The p-series test is a specific technique in the toolbox of Convergence Tests that focuses on series of the form \( \sum \frac{1}{k^p} \). It’s a straightforward method that helps quickly determine if a series converges or diverges by simply considering the exponent \( p \).

When using the p-series test, remember these rules:

• If \( p > 1 \), the series \( \sum \frac{1}{k^p} \) converges. That means the series will settle to a finite sum as \( k \) increases.
• If \( p \leq 1 \), the series diverges, which means it doesn’t settle to a finite sum.

In the given exercise, the series \( \sum \frac{1}{k^{7/6}} \) has \( p = 7/6 \), which is greater than 1. Therefore, according to the p-series test, this series converges.

Convergence Tests

Understanding and applying convergence tests is crucial for working with infinite series. These tests are like tools in a toolbox, each suited for different types of problems. Some tests are simple to apply, while others may require careful setup and analysis.

Common convergence tests include:

• Geometric Series Test: Determines convergence for series where each term is a fixed multiple of the preceding term. It converges if the common ratio’s absolute value is less than 1.
• Integral Test: Utilizes calculus to compare a series to a related continuous function.
• Ratio Test: Looks at the limit of the ratio of consecutive terms, providing information about absolute convergence.
• Root Test: Similar in spirit to the ratio test, it focuses on the nth root of the absolute value of terms.
• Limit Comparison Test: As discussed, compares to a known series to draw conclusions about convergence.
• p-series test: Specifically used for series in the form \( \sum \frac{1}{k^p} \).

Each test offers its advantages depending on the series at hand. By learning to choose the appropriate test, one can efficiently determine the behavior of infinite series, just like in the example provided. The combination of these tests allows you to tackle a wide range of convergence questions, ensuring mathematical rigor in your analysis.