Problem 71 - 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. $$\sum_{n=0}^{\infty} 5\left(-\frac{1}{2}\right)^{n}$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is \( \frac{10}{3} \).

Step 1: Identify the Common Ratio

In order to identify if this series is a geometric series, one needs to identify a common ratio between terms. In the given infinite series, the general term, denoted as \( a_{n} \), is \( 5 \left(-\frac{1}{2}\right)^{n} \). The existence of a power of \( n \) in the term confirms that it is indeed a geometric series. The base of that power, \( -\frac{1}{2} \), is the common ratio, \( r \).

Step 2: Check the Condition for Summation

To find the sum of an infinite geometric series, the value of the common ratio \( r \) must have an absolute value less than 1. In other words, \( |-1/2| < 1 \). It is clear that this condition is satisfied.

Step 3: Calculate the Sum

Since the ratio's absolute value is less than 1, the sum \( S \) of the infinite geometric series can be calculated using the following formula: \( S = \frac{a}{1 - r} \). Here \( a \) is the first term, and in our case, \( a = 5 \) because our series starts at \( n = 0 \). Hence, the sum can be calculated as \( S = \frac{5}{1 - (-1/2)} = \frac{5}{3/2} = \frac{10}{3} \).

Key Concepts

Common Ratio

In a geometric series, the common ratio is the constant factor between successive terms. It is crucial for determining whether a series is geometric. You multiply each term by the same number to get the next term, and this multiplier is the common ratio. It's a vital aspect since it helps identify the series' type and is used in the geometric series formula.

For instance, in the series given by the general term \( 5\left(-\frac{1}{2}\right)^{n} \), the common ratio \( r \) is \(-\frac{1}{2}\). Each term is obtained by multiplying the previous term by \( -\frac{1}{2} \).

To find the sum of an infinite geometric series, the common ratio must satisfy a specific condition. Specifically, the absolute value of the common ratio must be less than 1, which guarantees the terms are getting smaller in size and the series will converge to a sum. In our example, \( |-\frac{1}{2}| = \frac{1}{2} \) is less than 1, so the series converges.

Geometric Series Formula

The geometric series formula is used to find the sum of terms in a geometric series. For an infinite series, where terms continue indefinitely, the formula is applicable if the series converges. Convergence occurs when the common ratio's absolute value is less than one. This formula helps us quickly find the total of all terms without having to add each one individually.

The formula to find the sum \( S \) of an infinite geometric series is expressed as:

• \( S = \frac{a}{1 - r} \)

where \( a \) is the first term, and \( r \) is the common ratio. This formula relies on the fact that each successive term gets smaller and smaller, approaching zero, which makes the infinite sum possible.

In the example series \( \sum_{n=0}^{\infty} 5\left(-\frac{1}{2}\right)^{n} \), we use this formula. Here, \( a = 5 \), and \( r = -\frac{1}{2} \). Substituting these values gives the sum \( S = \frac{5}{1 - (-\frac{1}{2})} = \frac{5}{3/2} = \frac{10}{3} \). This elegant approach simplifies calculating the sum of infinite series.

Sum of Series

When it comes to infinite geometric series, not every series can be summed. The possibility of finding a sum comes down to whether the series converges, which is directly linked to the common ratio. If the series converges, it means adding infinite terms results in a finite number.

The sum of a convergent infinite geometric series is found using the geometric series formula. This formula provides a way to add up all the terms quickly and comprehensively even if they run infinitely.

In practice, this means if you have an initial term \( a \) and a common ratio \( r \) with \( |r| < 1 \), the series sum \( S \) is \( \frac{a}{1 - r} \). This method is powerful especially in real-world scenarios like calculating discounts, population growth, or interest rates where repeated multiplicative processes naturally form geometric series.

In our exercise, after determining that the series converges, the sum was calculated as \( \frac{10}{3} \). It shows how convergence and the geometric series formula work together to find a precise total from what initially seems to be an endless collection of numbers.