Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ \frac{2}{3}+2+6+\cdots+2(3)^{n}+\cdots $$

Step-by-Step Solution

Answer

The given series diverges and hence does not have a finite sum.

Step 1: Identify the Pattern

The first step is identifying if the series follows a specific pattern. Looking at the given series \( \frac{2}{3}, 2, 6, \ldots \), the pattern can be seen as each term being three times the previous term after the first term. This indicates a geometric series with first term \(a = \frac{2}{3}\) and common ratio \(r = 3\).

Step 2: Determine if the series Converges or Diverges

In a geometric series, a necessary condition for the series to converge is that the absolute value of the common ratio \(r\) is less than 1. However, in this case the common ratio is 3 which is greater than 1. Therefore, the given series will diverge.

Step 3: Sum the Series

Calculating the sum for a geometric series only makes sense if the series converges. Since we've just determined this series diverges, its sum is not a finite number.

Key Concepts

Convergence and Divergence

When examining series, particularly infinite ones like geometric series, it's crucial to determine whether they converge or diverge. **Convergence** simply means that as we sum more and more terms from the series, the total approaches a specific finite value. **Divergence** implies that the sum does not settle; it might grow indefinitely or not stabilize at any finite number.

• To test convergence in geometric series, the core factor is the common ratio, denoted as \( r \).
• If the absolute value of \( r \) is less than 1, \( |r| < 1 \), the series converges.
• Conversely, if \( |r| \geq 1 \), the series diverges.

In our original exercise, the common ratio \( r \) was 3, which led to divergence because it is greater than 1.

Common Ratio

The common ratio in a geometric series is a pivotal element that dictates the behavior of the series.Defined as the constant factor between successive terms, it is typically represented by \( r \).

• For example, consider the series \( \frac{2}{3} , 2, 6, \ldots \).
• Here, each term is calculated by multiplying the previous term by 3.

This multiplication by 3 establishes the common ratio as 3. To compute it in general, take any term in the series and divide it by the preceding term.If the terms are \( a, ar, ar^2, \ldots \), then the common ratio \( r = \frac{ar}{a} = r \).By understanding the common ratio, you can quickly assess whether a series will converge or diverge based on its value.

Infinite Series

An infinite series is a sum of an endless number of terms.This concept can be somewhat counterintuitive at first because although the number of terms seems boundless, the series might still have a finite sum if it converges.

• Infinite series can be derived from patterns observed in sequences.
• They expand endlessly in the form \( a, ar, ar^2, ar^3, \ldots \) for geometric types.

Understanding infinite series involves determining whether their total approaches a finite limit or not, as seen in the segment on convergence and divergence.In cases like the original exercise, where the series diverges, the infinite aspect means that no definite total sum exists and it stretches indefinitely.Through these insights, you can begin to see the interplay between finite concepts and infinite processes that create fascinating results in mathematics.