Problem 70 - Precalculus: Mathematics for Calculus

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$\frac{1}{3^{6}}+\frac{1}{3^{8}}+\frac{1}{3^{10}}+\frac{1}{3^{12}}+\cdots$$

Step-by-Step Solution

Answer

The series is convergent with a sum of \( \frac{1}{648} \).

Step 1: Identify the First Term and Common Ratio

The given series is \( \frac{1}{3^6} + \frac{1}{3^8} + \frac{1}{3^{10}} + \cdots \). Identify the first term \(a\) and the common ratio \(r\). Here, the first term \(a = \frac{1}{3^6}\) and the common ratio \(r\) is the ratio of any term to its preceding term, which is \( \frac{1}{3^8} \div \frac{1}{3^6} = \frac{1}{3^2} = \frac{1}{9} \).

Step 2: Determine Convergence or Divergence

A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. In our series, the common ratio \(r = \frac{1}{9}\), and since \(|\frac{1}{9}| < 1\), the series converges.

Step 3: Use the Formula for the Sum of a Convergent Series

For a convergent geometric series, the sum \(S\) can be calculated using the formula \( S = \frac{a}{1-r} \), where \(a\) is the first term and \(r\) is the common ratio. Substitute the values: \(a = \frac{1}{3^6}\) and \(r = \frac{1}{9}\).

Step 4: Calculate the Sum

Apply the values to the formula: \( S = \frac{\frac{1}{3^6}}{1 - \frac{1}{9}} = \frac{\frac{1}{729}}{\frac{8}{9}} = \frac{1}{729} \times \frac{9}{8} = \frac{1}{648} \). Therefore, the sum of the series is \( \frac{1}{648} \).

Key Concepts

Convergence

When we talk about convergence in the context of a geometric series, we're looking into whether the series approaches a finite sum as the number of terms grows indefinitely. A geometric series is a sum of the form \( a + ar + ar^2 + ar^3 + \ldots \), where \( a \) is the first term and \( r \) is the common ratio. The pivotal point to determine convergence is the value of \( r \).

• If \(|r| < 1\), the series converges to a finite value.
• If \(|r| \geq 1\), the series does not converge and the sum grows indefinitely.

To determine convergence, always examine the common ratio \( r \). If it falls under the threshold of 1 when taking its absolute value, you can confidently move on to calculating the sum.

Common Ratio

The common ratio \( r \) in a geometric series is a critical component which dictates the behavior of the series. It is calculated by dividing any term in the series by the previous term. For the series \( \frac{1}{3^6} + \frac{1}{3^8} + \frac{1}{3^{10}} + \cdots \), the common ratio is \( \frac{1}{9} \). Finding the common ratio helps you understand the growth or decay rate of the series.

• It tells us how each term relates to the previous one.
• In determining convergence, if \(|r|\) is less than 1, the series will approach a particular value, otherwise, it will not.

Understanding how to find and apply the common ratio is key to mastering the properties of geometric series.

Sum of a Series

For a convergent geometric series, the sum is the value it approaches as you keep adding more terms. The formula for the sum of an infinite geometric series is \( S = \frac{a}{1-r} \), where \( a \) is the first term, and \( r \) is the common ratio. In our example of the series \( \frac{1}{3^6} + \frac{1}{3^8} + \cdots \), the sum can be calculated once we establish that it converges.

• First, ensure \(|r| < 1\). In this case, \( r = \frac{1}{9} \), which satisfies the condition.
• Substitute \( a = \frac{1}{3^6} \) and \( r = \frac{1}{9} \) into the formula to find the sum.
• The solution yields \( S = \frac{1}{648} \), the finite sum of our converging series.

This formula is beneficial because it provides a simplistic way to find and confirm the sum of complex infinite geometric series.