Problem 12 - Algebra 2

Question

Decide whether each infinite geometric series diverges or converges. State whether each series has a sum. $$ 1+2+4+\ldots $$

Step-by-Step Solution

Answer

The given infinite geometric series diverges; thus, it does not have a sum.

Step 1: Identify the first term

The first term of the series is \( 1 \). This is the first number listed in the series.

Step 2: Identify the common ratio

The common ratio is determined by dividing a term by its preceding term. In this series, the second term \( 2 \) is divided by the first term \( 1 \), giving a common ratio of \( 2 \).

Step 3: Determine if the series converges or diverges based on the common ratio

As noted earlier, an infinite geometric series converges if the common ratio's absolute value is less than 1. In this case, the common ratio \( 2 \) is greater than 1, so the series diverges.

Step 4: Determine the sum of the series

The formula for the sum of an infinite geometric series can only be used when the series converges. Since this series diverges, it does not have a sum.

Key Concepts

Convergence and Divergence

When dealing with infinite geometric series, deciding whether a series converges or diverges is crucial. Convergence means that as you add more terms in the series, its sum approaches a specific finite value. On the flip side, divergence means that the series does not settle at a given number but continues to grow indefinitely.

For example, consider the series:

• 1 + 2 + 4 + \ldots

To determine convergence or divergence, examine the series' common ratio. If the absolute value of the common ratio is less than 1, the series converges. However, if the absolute value of the ratio is equal to or greater than 1, the series diverges. This means:

• Converges: \(|r| < 1\)
• Diverges: \(|r| \geq 1\)

For this particular series, since the common ratio \( r = 2 \) which is greater than 1, the series diverges.

Common Ratio

The common ratio is a core element of any geometric series. It determines how each term in the series relates to the previous one. To find the common ratio \( r \), divide any term in the series by the term that precedes it. In the series given, 1, 2, 4,..., each term is twice the one before it:

• \( \frac{2}{1} = 2 \)
• \( \frac{4}{2} = 2 \)

Within a geometric series, the behavior of the series tends to rely heavily on the common ratio as follows:

• If \(|r| < 1\), the terms get smaller and the series may converge.
• If \(|r| > 1\), the terms get larger and the series will likely diverge.
• If \(|r| = 1\), the series neither grows nor shrinks specifically and remains constant.

Thus, understanding the common ratio's value is essential to predicting the series' overall behavior.

Sum of a Series

Determining the sum of an infinite geometric series is possible only if the series converges. If a series converges, it means its sum or total value approaches a specific limit as the number of terms tends toward infinity. For a convergent series, the sum \( S \) is calculated using the formula:\[S = \frac{a}{1 - r}\]

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

For the given series 1 + 2 + 4 + \ldots, since the common ratio \( r = 2 \), and because \( \lvert 2 \rvert \geq 1 \), the series does not converge. Therefore, it does not have a finite sum.

By understanding the necessary conditions for using the sum formula, students can calculate the sum efficiently for geometric series that meet the convergence criteria. In our example, however, since it diverges, calculating the sum is not applicable.