Problem 25 - Essential Calculus Early Transcendentals

Question

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{k=1}^{\infty} k\left(\frac{2}{3}\right)^{k} $$

Step-by-Step Solution

Answer

The series is absolutely convergent.

Step 1: Test for Absolute Convergence

To test for absolute convergence, we consider the series \[ \sum_{k=1}^{\infty} \left| k\left(\frac{2}{3}\right)^{k} \right| \]which is the same as the original series because the original terms are already non-negative.

Step 2: Use the Ratio Test

Apply the Ratio Test to determine convergence. Consider:\[ \lim_{k \to \infty} \left| \frac{(k+1)\left(\frac{2}{3}\right)^{k+1}}{k\left(\frac{2}{3}\right)^{k}} \right| \]This simplifies to:\[ \lim_{k \to \infty} \frac{k+1}{k} \cdot \frac{2}{3} = \frac{2}{3} \cdot 1 = \frac{2}{3} \]Since \(\frac{2}{3} < 1\), the series is absolutely convergent by the Ratio Test.

Step 3: Conclusion

Since the series is absolutely convergent, there is no need to check for conditional convergence, as absolute convergence implies conditional convergence as well.

Key Concepts

Absolutely Convergent

An absolutely convergent series is a series where the sum of the absolute values of its terms also converges. To put it more simply, if you take each term of the series and make it positive (by taking its absolute value), and that new series converges, then the original series is said to be absolutely convergent.

The importance of absolute convergence lies in its reliability; an absolutely convergent series will converge no matter how you rearrange its terms. This makes it a stronger form of convergence compared to conditional convergence. For example, in the exercise, the series \( \sum_{k=1}^{\infty} k\left(\frac{2}{3}\right)^{k} \) was tested for absolute convergence. The series was determined as absolutely convergent since the absolute values of its terms, summed together, also converged.

If a series is absolutely convergent, it is also convergent in the ordinary sense—meaning that its sequence of partial sums has a finite limit.

Conditionally Convergent

Conditional convergence occurs with a series that converges, but it does not converge absolutely. This means that if you take the absolute value of each term in the series, and sum those, the series would diverge. However, without taking the absolute value, the series itself converges.

For example, the alternating harmonic series \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k} \) is a classic example of a conditionally convergent series. Its terms alternate in sign and decrease in magnitude without bound, making the series as a whole converge. Yet, if you remove the alternating sign and sum \( \sum_{k=1}^{\infty} \frac{1}{k} \), the series diverges, hence conditional convergence.

In the exercise, checking for conditional convergence wasn't necessary since absolute convergence was already established. Once absolute convergence is established, conditional convergence automatically follows.

Ratio Test

The Ratio Test is a powerful tool to determine whether a series is convergent or divergent. It is particularly useful for series where each term is of a form that involves factorials, exponentials, or powers, like the series in our exercise.

To apply the Ratio Test, you evaluate the limit:

• \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \), where \( a_k \) represents the terms of your series.
• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

In our exercise, to determine if the series \( \sum_{k=1}^{\infty} k\left(\frac{2}{3}\right)^{k} \) converges, the Ratio Test was applied, and the limit was calculated as \( \frac{2}{3} \). Since \( \frac{2}{3} < 1 \), the series was confirmed as absolutely convergent. This example shows how the Ratio Test simplifies the process of testing for convergence, providing a clear result for many types of series.