Question

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 2+\frac{4}{3}+\frac{8}{9}+\cdots $$

Short answer

The series converges and its sum is 6.

Step-by-step solution

- Identify the first term

The first term of the series is the first number in the sequence. For this series, the first term (denoted as \(a\)) is 2.

- Find the common ratio

The common ratio (denoted as \(r\)) is the ratio between any two consecutive terms. Divide the second term by the first term: \(\frac{\frac{4}{3}}{2} = \frac{4}{3} \times \frac{1}{2} = \frac{2}{3}\). Thus, the common ratio is \(\frac{2}{3}\).

- Determine the behavior of the series

An infinite geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. Since \(\frac{2}{3}\) is less than 1, the series converges.

- Use the sum formula for a converging geometric series

For a converging geometric series, the sum \(S\) can be found using the formula: \[ S = \frac{a}{1 - r} \]Here, \(a = 2\) and \(r = \frac{2}{3}\).

- Calculate the sum

Plug the values into the formula:\[ S = \frac{2}{1 - \frac{2}{3}} = \frac{2}{\frac{1}{3}} = 2 \times 3 = 6 \]Therefore, the sum of the series is 6.

Key concepts

Convergence of Series

An infinite geometric series can either converge or diverge. Convergence happens when the sum of an infinite number of terms approaches a specific value. For geometric series, convergence is determined by the absolute value of the common ratio, denoted as \(|r|\). If \(|r| < 1\), the series converges. Otherwise, it diverges if \(|r| \geq 1\). In this exercise, the common ratio \( \frac{2}{3} \) has an absolute value less than 1, so the series converges.
This means the infinite additions approach a finite sum.

Common Ratio

The common ratio is a key part of identifying geometric series. It stays the same for each pair of consecutive terms in the sequence. You find it by dividing any term by the previous term.
For example, if you divide the second term \( \frac{4}{3} \) by the first term \(2\), you get the common ratio \(\frac{4}{3} \times \frac{1}{2} = \frac{2}{3}\).
Understanding the common ratio is essential because it tells us whether the series converges or diverges and helps find the series' sum if it converges.

Sum of Series

For a converging series, we have a formula to find the sum. The sum \( S \) of an infinite geometric series with the first term \( a \) and common ratio \( r \) is given by:
\[ S = \frac{a}{1 - r} \]
In our example, the first term \( a \) is 2 and the common ratio \( r \) is \(\frac{2}{3}\). Plugging these into the formula: \[ S = \frac{2}{1 - \frac{2}{3}} = \frac{2}{\frac{1}{3}} = 2 \times 3 = 6 \]
Thus, the sum of this infinite series is 6.
This formula only works for series that converge, making the common ratio crucial for determining if we can find the sum.