Problem 16 - 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. \(\sum_{k=1}^{\infty} \frac{2}{3^{k}+1}\)

Step-by-Step Solution

Answer

The series \(\sum_{k=1}^{\infty} \frac{2}{3^{k}+1}\) converges.

Step 1 - Simplify the series

First, let's simplify the series and compare it with a series that we know whether it converges or diverges. The series is \(\sum_{k=1}^{\infty} \frac{2}{3^{k}+1}\). The denominator of each term can be simplified by observing that 3^k >> 1 for large k, making the '+1' insignificant. As a result, the series may be approximated by \(\sum_{k=1}^{\infty} \frac{2}{3^{k}}\).

Step 2 - Apply Geometric Series Test

We know that a geometric series \(\sum_{k=1}^{\infty} ar^{k-1}\) will converge if the common ratio r lies in the range -1 < r < 1. In our simplified series, it can be written in the form of a geometric series with \(a = 2/3\) and \(r = 1/3\). Hence, our simplified series is a convergent geometric series because its common ratio \(r = 1/3\) lies between -1 and 1.

Step 3 - Apply Comparison Test between original series and simplified series

From the comparison test, if 0 ≤ a_k ≤ b_k for all k and if the series \(\sum_{k=1}^{\infty} b_k\) converges, then so does \(\sum_{k=1}^{\infty} a_k\). From Step 1, we know that our original series \(\sum_{k=1}^{\infty} \frac{2}{3^{k}+1}\) is less than or equal to the simplified series \(\sum_{k=1}^{\infty} \frac{2}{3^{k}}\). From Step 2, the simplified series converges. So, using the comparison test, we can conclude that the original series \(\sum_{k=1}^{\infty} \frac{2}{3^{k}+1}\) also converges.

Key Concepts

Understanding Geometric Series Test

The Geometric Series Test is a helpful tool used to determine whether infinite series like \(\sum_{k=1}^{\infty} ar^{k-1} \) converge. In a geometric series:

• \(a\) is the first term.
• \(r\) is the common ratio.

For convergence, the absolute value of the common ratio must be less than 1, written as \(-1 < r < 1\). This condition ensures that as you progress through the series, the terms become exceedingly small and ultimately approach zero.

For example, in the simplified series from the exercise, \(\sum_{k=1}^{\infty} \frac{2}{3^{k}}\),

• \(a = 2/3\)
• \(r = 1/3\)

Since \(r = 1/3\) is clearly within the range \(-1 < r < 1\), we conclude that this geometric series converges. The Geometric Series Test is straightforward but powerful, saving us from complicated calculations when handled properly.

The Role of the Comparison Test

The Comparison Test is a technique used to determine the convergence or divergence of a series by comparing it to another series with known behavior. Here's how it works:

• Identify another series \(\sum b_k\) similar to the series \(\sum a_k\) you want to test.
• Ensure \(0 \leq a_k \leq b_k\) for all terms.
• If \(\sum b_k\) is convergent, then \(\sum a_k\) is also convergent.
• If \(\sum b_k\) is divergent, then \(\sum a_k\) is also divergent provided \(b_k \leq a_k\).

In the given problem, we use the series \(\sum_{k=1}^{\infty} \frac{2}{3^{k}}\), which converges, as our \(\sum b_k\). The original series \(\sum_{k=1}^{\infty} \frac{2}{3^{k}+1}\) is always less than \(\sum_{k=1}^{\infty} \frac{2}{3^{k}}\), which satisfies the conditions of the comparison test. Since our \(\sum b_k\) converges, our original \(\sum a_k\) also converges by the test. This approach is invaluable for series that don't easily fit standard tests directly.

Exploring Convergent Series

Convergent series are infinite series where the sum of the terms approaches a finite number. This concept is vital because it tells us whether a sum results in a meaningful, finite value or just grows indefinitely.

• For a series \(\sum_{k=1}^{\infty} a_k\) to converge, terms \(a_k\) must become small enough fast enough as \(k\) increases.
• Convergence is often determined using tests like the Geometric Series Test or the Comparison Test.
• Convergent series have many applications in calculus, physics, and engineering.

Within the exercise, the series \(\sum_{k=1}^{\infty} \frac{2}{3^{k}+1}\) was shown to converge through simplification to a comparable geometric series and application of the comparison test, demonstrating the practical use of these techniques. Understanding convergence helps you not only with individual problems but also in broader mathematical contexts.