Problem 8 - Calculus: Early Transcendentals

Question

Determine whether the series converges or diverges. \( \displaystyle \sum_{n = 1}^{\infty} \frac {6^n}{5^n - 1} \)

Step-by-Step Solution

Answer

The series diverges.

Step 1: Analyze the Series Structure

Observe that the given series is of the form \( \sum_{n = 1}^{\infty} \frac {6^n}{5^n - 1} \). This series contains terms that resemble a geometric series.

Step 2: Compare with a Geometric Series

Recognize that for large \( n \), \( \frac{6^n}{5^n - 1} \) closely approximates \( \frac{6^n}{5^n} = \left(\frac{6}{5}\right)^n \). A geometric series with \( r = \frac{6}{5} > 1 \) diverges.

Step 3: Apply the Limit Comparison Test

Use the Limit Comparison Test to determine convergence or divergence. Compare the given series with the geometric series \( \sum \left(\frac{6}{5}\right)^n \). Calculate \( \lim_{n \to \infty} \frac{\frac{6^n}{5^n - 1}}{\left(\frac{6}{5}\right)^n} \).

Step 4: Simplify the Limit Expression

Simplify the limit expression: \[ \lim_{n \to \infty} \frac{\frac{6^n}{5^n - 1}}{\left(\frac{6}{5}\right)^n} = \lim_{n \to \infty} \frac{6^n \cdot \left(\frac{5}{6}\right)^n}{5^n - 1} = \lim_{n \to \infty} \frac{5^n}{5^n - 1} \].

Step 5: Evaluate the Limit

Evaluate \( \lim_{n \to \infty} \frac{5^n}{5^n - 1} \). This limit approaches 1 as \( n \) becomes very large, since \( 5^n \) dominates over \(-1\).

Step 6: Conclusion of Limit Comparison Test

Since the limit is finite and non-zero, and the geometric series \( \sum \left(\frac{6}{5}\right)^n \) diverges, by the Limit Comparison Test, the original series \( \sum \frac{6^n}{5^n - 1} \) also diverges.

Key Concepts

Geometric Series

A geometric series is one where each term after the first is found by multiplying the previous one by a fixed, non-zero number. This number is called the common ratio. For example, in the series \( 3 + 3r + 3r^2 + 3r^3 + \ldots \), each term is obtained by multiplying 3 by \( r^n \), where \( n \) is the term index.

It is crucial to understand when a geometric series converges or diverges, as this will determine whether the sum approaches a finite value.

• If the common ratio \( r \) satisfies \(|r| < 1\), the series converges, and the sum of the series can be calculated as \( \frac{a}{1 - r} \), where \( a \) is the first term.
• If \(|r| \geq 1\), the series diverges, meaning it does not reach a finite sum as more terms are added.

In our original exercise, the series resembles \( \left( \frac{6}{5} \right)^n \), which has a ratio greater than 1, leading to divergence.

Limit Comparison Test

The Limit Comparison Test is a handy tool in mathematical analysis, particularly when dealing with series that are difficult to classify simply. By using this test, we can compare a complex series with another that has a known convergence behavior.

Here’s how it works:

• Identify two series you're interested in: the troublesome series \( \sum a_n \) and another convenient series \( \sum b_n \).
• Evaluate the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If this limit yields a positive, finite number, both series will either converge or diverge together.

In our exercise, the troublesome series ends up resembling \( \sum \left( \frac{6}{5} \right)^n \) for large \( n \), which is known to diverge. Since their limit comparison is a finite positive number, the original series also diverges.

Convergence and Divergence

Convergence and divergence are fundamental concepts when studying series in mathematical analysis. Understanding them helps determine if adding all terms of an infinite series results in a finite sum.

Convergence occurs when the sum of infinitely many terms approaches a specific value. This means that as more terms are added, they have less influence on the overall sum.

• An example is the series \( \sum \frac{1}{n^2} \), which converges.

Divergence, on the other hand, is when the sum of the infinite series does not settle towards any particular value. The series may grow without bound or oscillate indefinitely.

• A common example is \( \sum n \), which diverges as it grows without bound.

In our exercise, the original series diverges because it resembles a geometric series with a common ratio greater than one.

Mathematical Analysis

Mathematical analysis is a branch of mathematics that explores limits, continuity, differentiation, integration, measure, and infinite series. It sets the groundwork for further studies in these topics, providing deep insights into not only solving but also understanding problems.

Here’s what makes mathematical analysis essential:

• Rigorous Conceptual Framework: It relies on a precise approach to solving problems, ensuring results are logically sound.
• Application of Limits: Limits are foundational, particularly when dealing with infinite series like in our exercise.
• Clear Understanding of Functions: Understanding how functions behave at boundaries or over large domains is crucial.

The exercise dives into these concepts by using the Limit Comparison Test, showing the power of mathematical analysis to determine divergence of series such as \( \sum \frac{6^n}{5^n - 1} \). Through analysis, an abstract problem becomes a methodical process.