Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Find the sum of the infinite geometric series if it exists. $$2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is 3.

Step 1: Identify the first term and common ratio

The first term \( a \) of the series is \( 2 \). The common ratio \( r \) can be found by dividing the second term by the first term: \( r = \frac{2/3}{2} = \frac{1}{3} \).

Step 2: Check if the series is convergent

An infinite geometric series converges if the absolute value of the common ratio \( |r| \) is less than 1. In this case, \( r = \frac{1}{3} \), and \(|\frac{1}{3}| < 1\), so the series converges.

Step 3: Use the formula for the sum of an infinite geometric series

The sum \( S \) of an infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.

Step 4: Substitute the values into the formula

Substitute \( a = 2 \) and \( r = \frac{1}{3} \) into the sum formula: \[ S = \frac{2}{1 - \frac{1}{3}} \] Simplify the denominator: \( 1 - \frac{1}{3} = \frac{2}{3} \). Thus, \[ S = \frac{2}{\frac{2}{3}} \]

Step 5: Perform the division

To divide by a fraction, multiply by its reciprocal: \[ S = 2 \times \frac{3}{2} \] Simplify the expression by canceling the \(2\): \[ S = 3 \].

Key Concepts

Sum of Infinite Series

In mathematics, when we talk about the "sum of an infinite series," we are referring to the sum of all terms in a sequence, which could potentially have an infinite number of terms. Particularly, when dealing with geometric series, the concept involves adding terms that each decrease in size and effectively approach zero. The infinite series in our exercise is geometric because each term after the first is obtained by multiplying the previous term by the common ratio. If this ratio is a fraction less than one, the terms get smaller and the series can be summed.

In the given exercise:

• The sequence starts with 2 and continues with fractions like \(\frac{2}{3}\), \(\frac{2}{9}\), and so on.
• The first few terms rapidly decrease, which allows the series to be summed to a finite value, even though there are infinitely many terms.

The beauty of infinite geometric series is that they allow us to understand the sum of an endless progression that shrinks closer and closer to zero.

Geometric Series Convergence

For an infinite geometric series to have a sum, it must converge. Convergence in this context means that as you add more and more terms, the total sum approaches a specific number. This is only the case when the common ratio of the series is between \(-1\) and \(1\) (exclusive). If the ratio is larger than \(1\) or smaller than \(-1\), the series will not converge, as terms will either grow or oscillate without settling to a specific limit.

In our exercise:

• The common ratio \(r\) is \(\frac{1}{3}\), which satisfies \(|r| < 1|\). So, the series converges.
• This means as we add more terms, we approach a final sum rather than moving farther away.

Convergence is a crucial concept because it determines the possibility of finding a sum in infinite series, making it practical even when extending infinitely.

Infinite Series Formula

To find the sum of an infinite geometric series that converges, we utilize a simple yet powerful formula. This formula allows us to compute the sum directly using only the first term and the common ratio. For a series where the first term is \(a\) and the common ratio is \(r\), the sum \(S\) of the infinite series is calculated using:
\[ S = \frac{a}{1 - r} \]This formula is derived based on the idea that as you add more and more diminishing terms, they increasingly approach zero, reducing the overall sum impact beyond a certain point.

Applying the formula to the exercise:

• First term \(a\) is \(2\), and common ratio \(r\) is \(\frac{1}{3}\). Substitute these into the formula:
• \[ S = \frac{2}{1 - \frac{1}{3}} \]
• Solving gives us \( S = 3 \).

This formula is integral for solving many practical problems involving endless series, offering a graceful method to handle infinity.