Problem 51 - Algebra and Trigonometry

Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots $$

Step-by-Step Solution

Answer

The series is convergent with a sum of \( \frac{3}{2} \).

Step 1: Identify the Series

The given series is an infinite geometric series: \[ 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \]In a geometric series, each term is the product of the previous term and a constant ratio, known as the common ratio \( r \).

Step 2: Determine the First Term and Common Ratio

Identify the first term \( a \) and the common ratio \( r \):- The first term \( a = 1 \).- The common ratio \( r \) can be found by dividing the second term by the first term: \[ r = \frac{1/3}{1} = \frac{1}{3} \]

Step 3: Check for Convergence

A geometric series converges if the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \). Since \( r = \frac{1}{3} \), which is less than 1, the series converges.

Step 4: Calculate the Sum of the Series

For a convergent geometric series, the sum \( S \) can be calculated using the formula: \[ S = \frac{a}{1 - r} \]Substituting the values of \( a = 1 \) and \( r = \frac{1}{3} \), we get:\[ S = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

Key Concepts

Geometric Series

A geometric series is a fascinating mathematical concept where each term is derived by multiplying the previous term by a fixed constant, called the common ratio. This property makes the series unique and predictable in its progression.
For instance, consider the series in our example:

• The first term is 1.
• Each subsequent term is a product of the preceding term and the common ratio, which in this case is \( \frac{1}{3} \).

The sequence continues as follows: \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots\). Since the ratio is consistent across the series, it is classified as a geometric series. Understanding the geometric series helps in identifying the behavior of the infinite sequence and whether it will converge or diverge.

Infinite Series

Infinite series extend the concept of a sequence to the domain of infinity. In such series, terms continue indefinitely. There’s no last term here, and whether the series converges (approaches a specific number) or diverges (grows indefinitely or oscillates) is important.In an infinite geometric series, the characteristics and behavior depend largely on the common ratio. Specifically, an infinite geometric series converges if the absolute value of the common ratio is less than 1. For our series with a common ratio of \( \frac{1}{3} \), this means:

• The series converges because \( |\frac{1}{3}| < 1 \).
• This convergence implies that the series will sum to a finite value.

Knowing whether a series converges is critical in fields that rely heavily on calculations involving infinity, such as calculus and numerical analysis.

Common Ratio

The common ratio is the backbone of a geometric series. It's the number that each term in the series is multiplied by to get the next term. Determining this ratio is crucial for understanding the series itself.For example, in our series, the common ratio \( r \) is \( \frac{1}{3} \). This means:

• Each term is derived from the previous term by multiplying it by \( \frac{1}{3} \).
• This constant ratio determines the series' direction—whether it converges or diverges.

The condition for convergence of a geometric series is that the absolute value of the common ratio must be less than 1, \( |r| < 1 \). We determined that \( |\frac{1}{3}| = \frac{1}{3} \), satisfying the convergence condition, which directed us to calculate the finite sum of the series using the formula \( S = \frac{a}{1 - r} \). The common ratio, thus, is pivotal in the behavior and outcome of the series.