Problem 56 - Algebra and Trigonometry

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

The first term of the series is given as \(a = \frac{1}{3^6}\).

Step 2: Determine the common ratio

To determine the common ratio \(r\), divide the second term of the series by the first term: \(\frac{1}{3^8} \div \frac{1}{3^6} = \frac{1}{3^{8}} \times \frac{3^6}{1} = \frac{1}{3^2} = \frac{1}{9}.\) Therefore, \(r = \frac{1}{9}.\)

Step 3: Assess convergence of the series

An infinite geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. Here, \(|r| = \frac{1}{9} < 1\), so the series is convergent.

Step 4: Calculate the sum of the series

For a convergent geometric series, the sum \(S\) can be found using the formula \(S = \frac{a}{1 - r}\). Substitute \(a = \frac{1}{3^6}\) and \(r = \frac{1}{9}\): \[ S = \frac{\frac{1}{3^6}}{1 - \frac{1}{9}} = \frac{\frac{1}{729}}{\frac{8}{9}} = \frac{1}{729} \times \frac{9}{8} = \frac{9}{5832} = \frac{1}{648}. \]

Key Concepts

Convergence

The concept of convergence is crucial in understanding infinite geometric series. Convergence refers to the behavior of a series as it approaches a limit. In simpler terms, we say that an infinite series is convergent if the sum of its terms gets closer and closer to a specific value, rather than growing infinitely large or fluctuating. For a geometric series, we can determine its convergence by looking at the common ratio, denoted as \(r\). If the absolute value of \(r\) is less than 1, the series will converge. This means that as you add more and more terms together, the total sum of the series approaches a finite number. In the problem given, since the common ratio is \(\frac{1}{9}\), which is less than 1, the series is convergent. This informs us that we can find a finite sum for the infinite series.

Common Ratio

The common ratio \(r\) is a fundamental element of any geometric series. It signifies the constant factor between consecutive terms in the series. Knowing how to find and use the common ratio is essential for determining whether a series converges or diverges. In the series \(\frac{1}{3^6}+\frac{1}{3^8}+\frac{1}{3^{10}}+\cdots\), to find the common ratio, we divide the second term by the first term:

• Second term = \(\frac{1}{3^8}\)
• First term = \(\frac{1}{3^6}\)
• Common ratio \(r = \frac{\frac{1}{3^8}}{\frac{1}{3^6}} = \frac{1}{9}\)

The common ratio for this series is \(\frac{1}{9}\), indicating each term is \(\frac{1}{9}\) times the previous term.

Geometric Series Sum

When a geometric series is convergent, its sum can be calculated exactly, even if there are infinitely many terms. This is one of the key properties of convergent geometric series. The formula used to find the sum of an infinite geometric series 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. Notice that this formula applies only if \(|r| < 1\).With the series in the original exercise, the first term \(a = \frac{1}{3^6}\) and the common ratio \(r = \frac{1}{9}\). Plug these values into the formula:

• \(S = \frac{\frac{1}{3^6}}{1 - \frac{1}{9}}\)
• \(S = \frac{1/729}{8/9}\)
• \(S = \frac{9}{5832} = \frac{1}{648}\)

Thus, the sum of the infinite series is \(\frac{1}{648}\).

Infinite Series

An infinite series is a sum of infinitely many terms. While it might seem challenging to grasp the concept of adding up infinitely many numbers, infinite series have practical applications, particularly when they converge to a finite value. A geometric series, such as the one in this example, is a particular type of infinite series where each term is a multiple of the previous term, determined by the common ratio \(r\). Infinite series can either converge to a specific value or diverge, which means they do not reach a definite sum.The concept of infinite series plays an essential role in calculus and analysis. It enables mathematicians and scientists to solve problems related to limits, areas, and even in fields like signal processing and finance. Understanding when and how a series converges forms the basis of many advanced mathematical theories.