Problem 7 - Algebra 2

Question

Find the sum of each infinite geometric series, if it exists. $$\sum_{n=1}^{\infty} 40\left(\frac{3}{5}\right)^{n-1}$$

Step-by-Step Solution

Answer

The sum of the series is 100.

Step 1: Identify the First Term and Common Ratio

For the given infinite geometric series \( a_n = 40\left(\frac{3}{5}\right)^{n-1} \), the first term \( a_1 \) is the term when \( n = 1 \). Thus, \( a_1 = 40 \times \left(\frac{3}{5}\right)^0 = 40 \). The common ratio \( r \) is the factor by which we multiply each term to get the next term, which is \( \frac{3}{5} \). Thus, \( r = \frac{3}{5} \).

Step 2: Check Convergence of the Series

For an infinite geometric series to converge, the common ratio \( |r| \) must be less than 1. Here, \( |r| = \left|\frac{3}{5}\right| = \frac{3}{5} < 1 \), so the series converges.

Step 3: Apply the Geometric Series Formula

The sum \( S \) of an infinite geometric series is given by \( S = \frac{a_1}{1 - r} \), where \( a_1 \) is the first term and \( r \) is the common ratio. Here, \( a_1 = 40 \) and \( r = \frac{3}{5} \). So, \( S = \frac{40}{1 - \frac{3}{5}} = \frac{40}{\frac{2}{5}} \).

Step 4: Calculate the Sum

Now, calculate \( \frac{40}{\frac{2}{5}} \). This is the same as \( 40 \times \frac{5}{2} = 100 \). Therefore, the sum of the series is 100.

Key Concepts

Convergence

Convergence is a crucial concept when dealing with infinite geometric series. An infinite series converges when the sum approaches a specific value as the number of terms goes to infinity. But not all infinite series have a sum.
The convergence of an infinite geometric series is determined by its common ratio. Specifically, for convergence, the absolute value of the common ratio must be less than 1. This means that with each step, the added term becomes smaller and smaller, allowing the sum to stabilize at a particular value.
In our example, the common ratio is \( \frac{3}{5} \), and since \( \left| \frac{3}{5} \right| < 1 \), the series converges. This implies that as you add more terms, they'll have less impact on the total sum, leading to a finite and stable sum.
Understanding convergence ensures that we can find meaningful sums for infinite series, which otherwise would be undefined or infinite.

Common Ratio

The common ratio is a fundamental characteristic of a geometric series. It is the factor that you multiply by to move from one term of the series to the next. This is what determines the progression pattern of the series.
For a given series \( a_n = 40\left(\frac{3}{5}\right)^{n-1} \), the common ratio \( r \) is found by dividing any term by the previous term, and in this case, it is \( \frac{3}{5} \).

• A common ratio greater than 1 will cause the terms to increase.
• A common ratio less than 1, but greater than 0, will cause the terms to decrease.
• A negative common ratio will alternate the terms between positive and negative.

Understanding the role of the common ratio not only helps determine if the series converges but also provides insight into how the series behaves as you add more terms.

Geometric Series Formula

The geometric series formula is instrumental in finding the sum of an infinite series that converges. For series that meet the convergence criteria, the sum \( S \) can be calculated using the formula:
\[ S = \frac{a_1}{1 - r} \] Here, \( a_1 \) is the first term of the series, and \( r \) is the common ratio. In the example provided, \( a_1 = 40 \) and \( r = \frac{3}{5} \). By substituting these values into the formula:

• Compute the denominator: \( 1 - \frac{3}{5} = \frac{2}{5} \)
• Then, compute the sum: \( \frac{40}{\frac{2}{5}} = 40 \times \frac{5}{2} = 100 \)

This formula provides a quick path to the result, enabling you to bypass calculating each term individually. It's particularly powerful in accounting and financial contexts, where understanding the overall sum of an investment or loan is crucial.