Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series, if possible. If not possible, explain why. $$\sum_{n=0}^{\infty} 10\left(\frac{4}{5}\right)^{n}$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series \(\sum_{n=0}^{\infty} 10\left(\frac{4}{5}\right)^{n}\) is 50. If the absolute value of the common ratio had been greater than or equal to 1, the series would not have had a finite sum.

Step 1: Identify the parameters of the geometric series

Here, the first term of the series \(a = 10\) and the common ratio \(r = \frac{4}{5}\). Because the absolute value of the ratio \(|r| = \frac{4}{5} < 1\), the series is a convergent geometric series.

Step 2: Apply the formula for the sum of an infinite geometric series

The formula for the sum \(S\) of an infinite geometric series is given by \(S = \frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the common ratio.

Step 3: Calculate the sum

Substitute the given values into the formula: \(S = \frac{10}{1 - \frac{4}{5}} = \frac{10}{\frac{1}{5}} = 10 * 5 = 50\).

Key Concepts

Common Ratio

In an infinite geometric series, the common ratio is the factor that each term is multiplied by to get the next term. Identifying the common ratio, denoted as \( r \), is crucial for analyzing such series. For instance, in our series \( \sum_{n=0}^{\infty} 10\left(\frac{4}{5}\right)^{n} \), the common ratio \( r \) is \( \frac{4}{5} \). This fraction indicates that each subsequent term in the series is \( \frac{4}{5} \) of the previous term.

When dealing with the common ratio:

• If \(|r|\) is greater than 1, the terms get larger or more negative, and the series diverges.
• If \(|r|\) is less than 1, like in our example \(|r| = \frac{4}{5}\), the series' terms decrease in magnitude and head towards zero.

Understanding the common ratio helps predict the behavior of the series as it progresses.

Convergent Series

A convergent geometric series is one in which the sum of its infinite terms approaches a finite number. The series will only converge if its common ratio \( r \) satisfies the condition \(|r| < 1\). When this condition holds, the terms continuously decrease in size, allowing the infinite series to have a finite sum.

In the exercise provided, because \(|r| = \frac{4}{5} < 1\) holds true, the series is convergent. Such convergence means we can use specific formulas to find the sum of all the terms, even when the number of terms is infinite. Convergence provides us a powerful tool to make sense of infinite processes by relating them to finite concepts.

Sum Formula for Geometric Series

The sum of an infinite geometric series, if convergent, can be calculated using a simple formula. The formula is \( S = \frac{a}{1 - r} \), where \( a \) is the first term of the series, and \( r \) is the common ratio. This formula is derived from the series itself and takes advantage of the constant ratio between terms.

To determine the sum of our example series:

• First, identify that the first term \( a = 10 \).
• The common ratio \( r = \frac{4}{5} \).
• Then plug these values into the sum formula: \( S = \frac{10}{1 - \frac{4}{5}} = \frac{10}{\frac{1}{5}} = 50 \).

This calculation shows how even infinitely many diminishing terms can sum to a specific, finite number. Understanding this formula allows us to convert an abstract infinite process into a practical calculation that is easy to manage.