Problem 56 - 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, and its sum is \( \frac{1}{648} \).

Step 1: Identify Series Form

The given series is \( \frac{1}{3^6} + \frac{1}{3^8} + \frac{1}{3^{10}} + \frac{1}{3^{12}} + \cdots \). This is an infinite geometric series with the first term \( a = \frac{1}{3^6} \) and the common ratio \( r = \frac{1}{3^2} \).

Step 2: Determine Convergence

For an infinite geometric series \( a + ar + ar^2 + \cdots \) to be convergent, the common ratio \( r \) must satisfy \( |r| < 1 \). In this series, \( r = \frac{1}{3^2} \) or \( \frac{1}{9} \), which is less than 1. Thus, the series is convergent.

Step 3: Find the Sum of the Series

The sum \( S \) of an infinite convergent geometric series is given by \( S = \frac{a}{1 - r} \). Substituting the values \( a = \frac{1}{3^6} \) and \( r = \frac{1}{9} \), we compute:\[ S = \frac{\frac{1}{3^6}}{1 - \frac{1}{9}} = \frac{\frac{1}{729}}{\frac{8}{9}} \]Simplifying this:\[ S = \frac{1}{729} \times \frac{9}{8} = \frac{1}{648} \]Thus, the series converges to \( \frac{1}{648} \).

Key Concepts

Geometric Series Convergence

An infinite geometric series converges or diverges based on the value of its common ratio. To determine the convergence of a series like \( \frac{1}{3^6} + \frac{1}{3^8} + \frac{1}{3^{10}} + \cdots \), we need to carefully consider the common ratio.
For the series to converge, the absolute value of the common ratio \( r \) must be less than 1. This is denoted by the condition \(|r| < 1\). If this condition is met, the series will converge, and it means the terms are getting smaller and approaching zero as you sum them up.
In our series, the common ratio \( r \) is \( \frac{1}{9} \), which is less than 1. This indicates that our series is convergent.

• If \(|r| < 1\): Series converges.
• If \(|r| \geq 1\): Series diverges.

Identifying convergence is the first crucial step in working with infinite geometric series, as it tells us whether we can proceed to find a finite sum.

Sum of Geometric Series

Once we establish that a geometric series is convergent, we can find its sum. The formula for the sum \( S \) of an infinite geometric series is given by \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.
For our series \( \frac{1}{3^6} + \frac{1}{3^8} + \frac{1}{3^{10}} + \cdots \), the first term \( a \) is \( \frac{1}{729} \) and the common ratio \( r \) is \( \frac{1}{9} \).
We substitute these values into the formula:\[ S = \frac{\frac{1}{3^6}}{1 - \frac{1}{9}} = \frac{\frac{1}{729}}{\frac{8}{9}} \]Upon simplifying the expression, the sum of the series is:\[ S = \frac{1}{729} \times \frac{9}{8} = \frac{1}{648} \]This indicates that the sum of the infinite series is \( \frac{1}{648} \). Finding the sum is powerful because it transforms an infinite series into a finite, comprehensible value.

Common Ratio in Series

The common ratio in a geometric series is the factor by which we multiply each term to get to the next term. It is a constant ratio, marked as \( r \), between consecutive terms and holds a key role in determining the nature of the series.
For our specific series, each term is derived by multiplying the previous term by \( \frac{1}{9} \). This ratio makes our series a classic example of an infinite geometric series with a decreasing pattern.
To find the common ratio, observe the following steps:

• Identify two consecutive terms in the series.
• Divide a term by its preceding term.

In our setup:- The second term \( \frac{1}{3^8} \) divided by the first term \( \frac{1}{3^6} \) gives \( \frac{1}{3^2} = \frac{1}{9} \).
Understanding the common ratio helps in identifying series behavior:

• If \( |r| < 1 \): Series terms get smaller over time, suggesting convergence.
• If \( |r| \geq 1 \): Series terms do not decrease to zero, indicating divergence.

Thus, identifying and understanding the common ratio provides insight into the series' evolution and eventual convergence or divergence.