Problem 47 - College Algebra with Modeling and Visualization

Question

Find the sum of the infinite geometric series. $$ 6-4+\frac{8}{3}-\frac{16}{9}+\frac{32}{27}-\frac{64}{81}+\cdots $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is \( \frac{18}{5} \).

Step 1: Identify the First Term

The first term of the series is the very first number in the series. By looking at the series, the first term \( a \) is 6.

Step 2: Find the Common Ratio

To find the common ratio \( r \) of the series, divide the second term by the first term. \[ r = \frac{-4}{6} = -\frac{2}{3} \] Check this ratio with another pair of consecutive terms to confirm it's consistent. The third term divided by the second term is \[ \frac{8/3}{-4} = -\frac{2}{3} \], which confirms the common ratio is \( r = -\frac{2}{3} \).

Step 3: Use the Infinite Series Sum Formula

For an infinite geometric series where \(|r| < 1\), the sum \( S \) can be calculated using the formula: \[ S = \frac{a}{1 - r} \] Substitute the first term \( a = 6 \) and common ratio \( r = -\frac{2}{3} \) into the formula.

Step 4: Calculate the Sum

Substitute the values into the formula: \[ S = \frac{6}{1 - (-\frac{2}{3})} = \frac{6}{1 + \frac{2}{3}} = \frac{6}{\frac{5}{3}} \] Simplify the expression: \[ S = 6 \times \frac{3}{5} = \frac{18}{5} \] Therefore, the sum of the series is \( \frac{18}{5} \).

Key Concepts

Sum of Infinite Series

When dealing with infinite geometric series, one might initially wonder if it's even possible to calculate a sum for an infinite amount of numbers or terms. The answer is yes, but only under certain conditions. An infinite geometric series can have a finite sum if its common ratio – the value we multiply each term by to get the next – is less than 1 in absolute value.

The sum of an infinite geometric series is calculated using a specific formula. This formula allows us to determine the total of all terms, even if they never end by themselves. For any infinite geometric series where the absolute value of the common ratio is less than 1, the sum \( S \) is found using the equation:

• \[ S = \frac{a}{1 - r} \]

Where \( a \) is the first term of the series and \( r \) is the common ratio. The formula makes it straightforward to sum an endless list of numbers, simplifying what could be an overwhelming task.

Geometric Series Formula

The geometric series formula is your key tool for resolving problems involving geometric series, whether infinite or finite. In the case of an infinite geometric series, the formula mentioned above helps find the sum when you have an initial term and a consistent ratio between terms.

To apply the formula effectively:

• Identify the first term, commonly denoted as \( a \).
• Determine the common ratio, the factor by which you multiply subsequent terms.
• Check that the absolute value of the common ratio is less than 1, which is necessary for the series to have a sum.
• Substitute these values into the sum formula.

By calculating using this formula, you capture the entire series' sum with symmetry and simplicity. It breaks down a potentially infinite process into a manageable calculation, leveraging mathematical principles to identify patterns and predict outcomes effectively.

Common Ratio

The concept of the common ratio is central to understanding geometric series. This is the consistent factor that each term in the series is multiplied by to get the next term. Identifying it accurately is crucial in ensuring all subsequent calculations, especially the sum of the series, are correct.

To find the common ratio \( r \), simply divide one term by its preceding term, provided \( r \) remains consistent throughout the series:

• Example: In the series \(6, -4, \frac{8}{3} \dots \), dividing -4 (the second term) by 6 (first term) yields \( r = -\frac{2}{3} \).
• Verify by checking other terms, like dividing \( \frac{8}{3} \) by -4, confirming \( r = -\frac{2}{3} \) again.

It’s important to ensure that this common ratio remains the same for each pair of consecutive terms in the series. This consistency allows the application of the geometric series formula, which depends on \( r \), and ensures you can reliably predict the behavior of the series and its sum.