Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}-1}\)

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identifying the ratio

Identify the ratio in the geometric sequence. Here the ratio is \( \frac{3}{2} \).

Step 2: Check the condition for convergence

Check whether the magnitude of the ratio is less than 1. If the absolute value of ratio \( \frac{3}{2} \) is less than 1, the series will converge. If the absolute value is larger than or equal to 1, the series diverges.

Step 3: Conclusion

The ratio, \( \frac{3}{2} > 1 \), so it follows that the series \(\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}-1}\) diverges.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the ratio. For the series to converge, which means to approach a finite limit, this ratio must be between -1 and 1. To identify whether a series is geometric, you can divide successive terms and if the result is a constant, you're dealing with a geometric series.

For example, in the series \(\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}-1}\), we might initially suspect it to be geometric because each term involves an exponent ratio. In a pure geometric series, you'd see a form like \(\sum_{n=0}^{\infty} ar^n\), where \(a\) is the first term and \(r\) is the common ratio. Here, the problem is slightly more complex due to the \(2^{n}-1\) in the denominator which prevents it from being a conventional geometric series. However, understanding geometric series sets a foundation for analyzing series that exhibit similar patterns.

Ratio Test for Convergence

The ratio test for convergence is a powerful tool for determining whether a series converges or diverges. To apply the ratio test, take the limit of the absolute value of the ratio of consecutive terms, that is, \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).

If the limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges. When the limit is exactly 1, the test is inconclusive. For the given series \(\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}-1}\), the Ratio Test might seem applicable at a glance. However, since the conditions include the fact that all terms \(a_n\) and \(a_{n+1}\) should be positive and should not be affected by a non-constant term, as in the \(2^{n}-1\) part here, the Ratio Test would not be strictly appropriate for this series. Instead, a comparison test can be used to show that since \(\frac{3}{2} > 1\), we would expect divergence.

Divergence of Series

Understanding divergence is just as important as understanding convergence. A series diverges if it does not converge, meaning that the sum of its terms does not approach a finite limit as n increases indefinitely. In other words, the series either increases without bound or oscillates indefinitely without settling to a limit.

The given series \(\sum_{n=1}^{\infty} \frac{3^{n}}{2^{n}-1}\) is an example, where the divergence is evident from the ratio \(\frac{3}{2}\) derived from the sequence terms, which is greater than 1. It's important to note that divergence doesn’t always mean that the sequence's terms become infinitely large. Some divergent series can have terms that approach zero, yet their sum still does not converge to a finite number. This emphasizes the fact that analyzing the behavior of the individual terms is crucial but not sufficient; instead, we must consider the sum of the entire series.