Problem 25 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 25 through 32, determine whether the series converges or diverges. In this set of problems knowledge of the Limit Comparison Test is assumed. \(\sum_{k=1}^{\infty} \frac{3}{2^{k}-1}\)

Step-by-Step Solution

Answer

The given series \(\sum_{k=1}^{\infty} \frac{3}{2^{k}-1}\) converges using the Limit Comparison Test.

Step 1: Set Up the Test

Begin by setting up the Limit Comparison Test. Using this test, pick a series that is known to converge or diverge. In this case, we use \(b_k = \frac{1}{2^k}\), which converges because it's a geometric series where the ratio \(\frac{1}{2} < 1\).

Step 2: Calculate the Limit

Next, calculate the limit of the ratio \(\frac{a_k}{b_k}\) as \(k\) approaches infinity. So, we will find \(\lim_{k \to \infty} \frac{\frac{3}{2^k - 1}}{\frac{1}{2^k}}\). After simplification, this gives \(\lim_{k \to \infty} \frac{3 \times 2^k}{2^k - 1}\)

Step 3: Further Simplify and Evaluate the Limit

As \(k\) approaches infinity, the highest power of \(k\) in the denominator and the numerator take control, thus we can simplify our expression to \(\lim_{k \to \infty} \frac{3 \times 2^k}{2^k}\). This simplifies to \(\lim_{k \to \infty} 3\), which gives a finite constant, 3.

Step 4: Check the Resulting Limit

Since the limit computed is a finite positive number and the series we compared to (the one with terms \(b_k = \frac{1}{2^k}\)) converges, we can conclude that the original series \(\sum_{k=1}^{\infty} \frac{3}{2^k - 1}\) also converges.

Key Concepts

Series Convergence

Understanding series convergence is vital for determining whether a series sums to a finite value or not. An infinite series convergence means that as you keep summing the terms, the series approaches a finite number rather than blowing up to infinity.

To analyze any series,

• ensure that the terms are trending towards zero as the number of terms increases.
• Apply convergence tests like the Limit Comparison Test to decide if a series converges or diverges.

The value these tests find helps determine if adding infinitely many terms results in a finite sum. If the conditions of the chosen test are satisfied, we can confidently state whether the series converges. Central to our discussion is the fact that a converging series allows its terms to decrease in size fast enough to outweigh the infinite number of terms being added together.

If you understand that convergence indicates a kind of 'balance' in infinite additions, you'll find it easier to work through more complex problems!

Geometric Series

A geometric series is a special type of series where each term is a constant multiple of the previous term, defined by its common ratio \(r\). For example, the series \[ \sum_{n=0}^{fty} ar^n \] consists of terms like \(a, ar, ar^2, \ldots\) . Geometric series follow this easily-recognized style, making them a popular choice in comparison tests.

Some key points about geometric series are:

• If \(|r| < 1\), the geometric series converges to \( \frac{a}{1-r} \).
• If \(|r| \geq 1\), the series diverges.

To illustrate, in our exercise we compared our unknown series with the geometric series \(b_k = \frac{1}{2^k}\). Since the common ratio \(\frac{1}{2}\) is less than 1, this series converges rapidly, proving useful for our comparison test.

Understanding geometric series simplifies many problems similar to this one, allowing us to immediately know the convergence status without complex calculations.

Infinite Series

An infinite series forms the basis for understanding how sums work when they extend endlessly. When trying to determine characteristics of infinite series, you need to evaluate whether the series converges or not.

Considerations include:

• The nature and behavior of the series' terms as they approach infinity.
• Whether a series adds up to a specific value, becomes infinitely large, or oscillates.

In mathematical notation, infinite series are written as \( \sum_{k=1}^{\infty} a_k \), where variables \(a_k\) represent individual terms.

By applying methods like the Limit Comparison Test mentioned in our original problem, we impose structure onto the chaos of infinite additions. Such structured approaches allow us to definitively say a series converges when its terms add up constructively or diverges when they do not. This understanding of infinite series is critical not just in pure mathematics, but in understanding phenomena like signal processing or financial models.