Problem 8 - Applied Calculus

Question

Find the sum, if it exists. $$ 500(0.4)+500(0.4)^{2}+500(0.4)^{3}+\cdots $$

Step-by-Step Solution

Answer

The sum of the series is approximately 333.33.

Step 1: Identify the Series

The given series is an infinite geometric series: \[ 500(0.4) + 500(0.4)^2 + 500(0.4)^3 + ext{...} \] This can be rewritten in the standard form for a geometric series as \[ 500 imes (0.4)^n, \] where \( n \geq 1 \).

Step 2: Define the Components

Identify the first term \( a \) and the common ratio \( r \) of the geometric series: - The first term \( a = 500(0.4) = 200 \) - The common ratio \( r = 0.4 \)

Step 3: Check for Convergence

A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). In this case, \( |0.4| < 1 \), so the series converges.

Step 4: Calculate the Sum

Use the formula for the sum of an infinite geometric series, \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Substitute the values: \[ S = \frac{200}{1 - 0.4} = \frac{200}{0.6} \] Calculate further: \[ S = \frac{200}{0.6} = \frac{2000}{6} \approx 333.33 \]

Key Concepts

Convergence

In the context of infinite geometric series, convergence signifies whether the series approaches a certain value as the number of terms grows indefinitely. To determine convergence, we examine the absolute value of the common ratio, which is a key factor in assessing if the sum will stabilize at a finite number.
The general rule for convergence in an infinite geometric series is straightforward: the series will converge if the absolute value of the common ratio \( |r| \) is less than 1. This condition ensures that as the terms continue to be added, they become smaller and smaller, eventually allowing the overall sum to reach a finite limit.
In our example, the common ratio \( r \) is 0.4, and since \( |0.4| < 1 \), the series converges. This means that by adding more terms, the series doesn't grow indefinitely, but rather, it approaches a specific sum.

Sum of Series

The sum of an infinite geometric series with a common ratio that fulfills the convergence condition (i.e., \( |r| < 1 \)) can be calculated using a special formula. This formula is:
\[ S = \frac{a}{1 - r} \] where \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio. This formula helps us find a precise value for the sum, demonstrating the power of convergence.
In the given series, the first term \( a \) is 200, and the common ratio \( r \) is 0.4. By using the formula:

• Plug in the values: \( S = \frac{200}{1 - 0.4} \)
• Compute the expression: \( S = \frac{200}{0.6} \)
• Simplify to find the total sum: \( S = \frac{2000}{6} \approx 333.33 \)

This solution shows that the sum of this infinite geometric series is around 333.33 when rounded to two decimal places.

Common Ratio

The common ratio in a geometric series is a crucial component that defines how each term in the series relates to the one before it. Essentially, it's the factor by which you multiply a term to get the next term in the sequence. This ratio remains constant throughout the series, which is why it's called the "common" ratio.
In an infinite geometric series like the one presented, identifying the common ratio helps determine several important properties, including the potential convergence and the sum of the series.
For the given series:

• The common ratio is found by dividing the second term by the first term, though here it is provided as \( r = 0.4 \).
• This value of 0.4 means that each successive term is 40% of the one before it. Therefore, every time you move from one term to the next, you multiply the previous term by 0.4.
• The small size of the common ratio (less than 1) is exactly why this series converges instead of diverging, ultimately allowing us to calculate its sum.

Understanding the common ratio is essential for evaluating the behaviour of both finite and infinite geometric series.