Problem 79 - 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. $$9+6+4+\frac{8}{3}+\cdots$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series \(9 + 6 + 4 + \frac{8}{3} + \cdots\) is 27.

Step 1: Identify the common ratio

The first task is to identify the common ratio, which is the factor by which each term is multiplied to obtain the next term. This can be determined by dividing a term in the series by the preceding term. So, \[ r = \frac{6}{9} = \frac{2}{3} \].

Step 2: Validate the common ratio

Verify that the common ratio applies to all subsequent terms, by checking if each term can be obtained by multiplying the previous term by the common ratio. For example, the third term is \(4 = 6*\frac{2}{3}\), the fourth term is \(\frac{8}{3} = 4*\frac{2}{3}\) and so forth. This confirms the common ratio.

Step 3: Apply the formula for infinite geometric series sum

Apply the formula to compute the sum of an infinite geometric series. The sum, S, is calculated using the formula: \[ S = \frac{a}{1-r} \] where 'a' is the first term in the series (\(a=9\) in this case) and 'r' is the common ratio (\(r=\frac{2}{3}\)). Using these values, the sum, S, of the given infinite geometric series is: \[ S = \frac{9}{1-\frac{2}{3}} = \frac{9}{\frac{1}{3}} = 9*3 = 27 \]. This calculation is only valid because the absolute value of the common ratio (\(\left|\frac{2}{3}\right|\)) is less than 1. If this was not the case, the series would not have a sum.

Step 4: Interpret the result

The computed result represents the value towards which the cumulative sum of the terms in the series would approach if we were to carry out the additions indefinitely. In this case, if we keep adding subsequent terms in the given series, the summation will tend towards 27.

Key Concepts

Common Ratio

In a geometric series, the common ratio is a crucial element that connects each term in the series. It is the constant factor by which you multiply a term to get the next term. To find the common ratio, divide any term in the sequence by the previous term. For the given series \(9, 6, 4, \frac{8}{3}, \ldots\), we calculate the common ratio \(r\) as follows: \(r = \frac{6}{9} = \frac{2}{3}\).
This means that each term is multiplied by \(\frac{2}{3}\) to get the next one. It’s essential to verify this ratio for more than just the first pair of terms to ensure it's consistent throughout the series. For instance, \(4 = 6 \times \frac{2}{3}\) and \(\frac{8}{3} = 4 \times \frac{2}{3}\), showing that the common ratio of \(\frac{2}{3}\) holds true.

Sum of Infinite Series

The sum of an infinite geometric series is a fascinating concept! If the series has a common ratio with an absolute value less than 1, it converges to a finite sum. This sum can be found using a specific formula. In our case, the series \(9 + 6 + 4 + \frac{8}{3} + \ldots \) has a first term \(a = 9\) and a common ratio \(r = \frac{2}{3}\).
Because \(|r| < 1\), we can calculate the sum using the formula for infinite series. The sum \(S\) of such a series is \( S = \frac{a}{1-r} \).
Calculating, we get: \[ S = \frac{9}{1 - \frac{2}{3}} = \frac{9}{\frac{1}{3}} = 27 \].
This means that as we continue to add terms indefinitely, the series approaches a total sum of 27.

Geometric Series Formula

The geometric series formula provides the means to calculate the sum of terms in a geometric series. When dealing with an infinite series and the common ratio \(r\) satisfies \(|r| < 1\), the formula \(S = \frac{a}{1-r}\) is used, where \(a\) is the first term.
This formula is derived from understanding how the terms scale with each iteration. Each term is essentially shrinking by the factor of the common ratio \(r\). For example, in the series \(9, 6, 4, \ldots\), each term reduces relative to its previous term by the ratio \(\frac{2}{3}\).
The formula captures this shrinking relationship and allows for the calculation of the total sum of an infinite series by effectively adding diminishing amounts of the original term.

Convergence of Series

The concept of convergence in series is about determining whether the sum of a series approaches a specific value as more terms are added. It's a crucial idea, especially for infinite series. For geometric series, convergence depends heavily on the common ratio.
If the common ratio \(r\) is such that \(|r| < 1\), the series converges. This happens because each subsequent term becomes smaller, eventually nearing zero, leaving the sum to stabilize at a particular value.
In our given series \(9, 6, 4, \frac{8}{3}, \ldots\), the common ratio \(\frac{2}{3}\) ensures that the series converges. With \(|r| < 1\), the sum stops growing indefinitely and approaches 27 as more terms are added. If \(|r| \geq 1\), the series would diverge, meaning it wouldn't settle at a single sum.