Problem 47 - Precalculus: Mathematical for Calculus

Question

Find the sum of the infinite geometric series. $$1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{3}{2} \).

Step 1: Identify the First Term

Identify the first term of the series, which is the first number in the sequence. For this series, the first term is 1.

Step 2: Identify the Common Ratio

Determine the common ratio (r) by dividing the second term by the first term. Here, the second term is \( \frac{1}{3} \) and the first term is 1, so the common ratio is \( r = \frac{1}{3} \).

Step 3: Check the Sum Formula's Applicability

Remember that the sum of an infinite geometric series \( S \) is given by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio, provided \(|r| < 1\). Since \( |r| = \frac{1}{3} < 1 \), the formula can be applied.

Step 4: Apply the Sum Formula

Substitute the identified values into the formula for the sum of the series: \( S = \frac{1}{1 - \frac{1}{3}} \).

Step 5: Simplify the Expression

Simplify the expression by computing the denominator: \( 1 - \frac{1}{3} = \frac{2}{3} \), so \( S = \frac{1}{\frac{2}{3}} \).

Step 6: Compute the Sum

Simplify the fraction further by multiplying by the reciprocal of the denominator: \( S = 1 \times \frac{3}{2} = \frac{3}{2} \).

Step 7: Final Step: State the Result

Therefore, the sum of the infinite geometric series is \( \frac{3}{2} \).

Key Concepts

Common Ratio

In the context of an infinite geometric series, understanding the concept of the common ratio is crucial. The common ratio, denoted by \( r \), is the factor by which each term in the series is multiplied to obtain the next term. To find the common ratio, divide any term by the preceding term. For example, in the series you provided, the second term is \( \frac{1}{3} \) and the first term is 1. So, the common ratio \( r \) is \( \frac{1}{3} \).

• If \( r > 1 \) or \( r < -1 \), the terms will grow larger in magnitude, causing the series to diverge.
• If \( 0 < |r| < 1 \), the terms get smaller, enabling convergence to a sum.

In your series, the common ratio \( \frac{1}{3} \) ensures convergence because \( |r| < 1 \), allowing for a specific sum when using the sum formula.

Sum Formula

To determine the sum of an infinite geometric series, we use the sum formula. For any infinite geometric series where \(|r| < 1\), this sum can be calculated using:\[ S = \frac{a}{1 - r} \]Here, \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio.

• The condition \(|r| < 1\) ensures that the series converges, making it possible to find a finite sum.
• If \( |r| \geq 1 \), the series does not converge, and so does not have a sum in the conventional sense.

In this particular series, the sum formula applies because the common ratio is \( \frac{1}{3} \). Substituting the values \( a = 1 \) and \( r = \frac{1}{3} \) into the formula simplifies to \( S = \frac{1}{1 - \frac{1}{3}} = \frac{3}{2} \). This calculation reveals that the sum of this series is \( \frac{3}{2} \).

First Term

Identifying the first term, \( a \), of a geometric series establishes the starting point of the sequence. This term is crucial not only in determining the progression of the series but also in calculating the sum when using the sum formula.In your example, the first term \( a \) is 1, which is evident as it is the initial value in the sequence \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots\).

• The first term is vital in the sum formula \( S = \frac{a}{1 - r} \) as it directly influences the result of the series' sum.
• A different first term fundamentally changes the sequence and thus the sum of the series.

Remember, the series' configuration begins with this term, which sets up the entire series' behavior and sum. By knowing the first term and common ratio, one can determine any term in the series using the formula \( a_n = a \cdot r^{n-1} \). This capability makes the first term a cornerstone of analyzing geometric series.