Problem 44 - College Algebra

Question

Find the sum of the infinite geometric series. $$ \sum_{\infty}^{k=1} 3 \cdot\left(\frac{1}{4}\right)^{k-1} $$

Step-by-Step Solution

Answer

The sum of the series is 4.

Step 1: Identify the First Term

To find the sum of an infinite geometric series, we first identify the first term \( a \). In the series \( \sum_{k=1}^{\infty} 3 \cdot \left(\frac{1}{4}\right)^{k-1} \), the expression when \( k = 1 \) is the first term. Therefore, \( a = 3 \cdot \left(\frac{1}{4}\right)^{1-1} = 3 \cdot 1 = 3 \).

Step 2: Determine the Common Ratio

Next, we identify the common ratio \( r \) of the series. The general form of each term after the first is \( 3 \cdot \left(\frac{1}{4}\right)^{k-1} \). The common ratio is the base that the exponent \((k-1)\) affects, which is \( \frac{1}{4} \). Thus, \( r = \frac{1}{4} \).

Step 3: Confirm the Series is Convergent

For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1, \( |r| < 1 \). Since \( |\frac{1}{4}| = \frac{1}{4} \), which is less than 1, the series converges.

Step 4: Apply the Sum Formula for an Infinite Series

The sum \( S \) of an infinite geometric series where \( |r| < 1 \) is given by the formula \( S = \frac{a}{1 - r} \). Substitute \( a = 3 \) and \( r = \frac{1}{4} \) into the formula: \[ S = \frac{3}{1 - \frac{1}{4}} = \frac{3}{\frac{3}{4}} = 3 \times \frac{4}{3} = 4. \]

Step 5: State the Sum of the Series

The calculated sum of the infinite geometric series \( \sum_{k=1}^{\infty} 3 \cdot \left(\frac{1}{4}\right)^{k-1} \) is 4.

Key Concepts

Geometric Series Sum

A geometric series is a series of numbers where each term is a fixed multiple of the previous term. This fixed multiple is called the **common ratio**. For an infinite geometric series, finding the sum might seem daunting, but it is quite manageable with a specific formula. If an infinite geometric series converges, its sum can be calculated using the formula:

• \( S = \frac{a}{1 - r} \)

Here, \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio. Notably, this formula is only applicable when the absolute value of \( r \) is less than 1.
In the provided example, the first term \( a \) is 3, and the common ratio \( r \) is \( \frac{1}{4} \). Substituting these values into the formula gives us the sum of 4 for the infinite series:

• \( S = \frac{3}{1 - \frac{1}{4}} = 4 \)

Understanding this formula saves much effort and provides an elegant solution to what might seem infinite.

Common Ratio

The common ratio \( r \) in a geometric series is the factor by which each term is multiplied to produce the next term. It is pivotal in determining both the pattern of the series and whether the series converges or diverges. In any geometric series, the common ratio can be found by dividing any term by its preceding term. In mathematical notation, the common ratio can be expressed as:

• \( r = \frac{a_{n}}{a_{n-1}} \)

This ratio remains constant throughout the series. For example, in our infinite series \( 3 \cdot \left(\frac{1}{4}\right)^k \), the common ratio is \( \frac{1}{4} \). This value of \( r \) plays a crucial role in determining if the series sum can be calculated using the formula for convergent series.

Series Convergence

A series is said to be convergent when its terms approach a specific finite value as the number of terms grows infinitely large. For infinite geometric series, convergence is determined by the size of the common ratio \( r \).To decide if a geometric series like ours converges, you check whether the absolute value of the common ratio is less than 1:

• \( |r| < 1 \)

If this condition holds true, the series converges, meaning it approaches a specific sum. In the given problem, \( r = \frac{1}{4} \) and since \( \left| \frac{1}{4} \right| = \frac{1}{4} < 1 \), the series converges.
This concept is essential as it ensures the formula for the geometric series sum can be applied. Without convergence, the series would not approach a sum, making the series have no real, meaningful sum.