Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, 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 (a)

The first term of the series is the value of the series when \( k = 1 \). Since the formula for each term is given as \( 3 \cdot \left( \frac{1}{4} \right)^{k-1} \), substituting \( k = 1 \) gives us the first term \( a = 3 \cdot \left( \frac{1}{4} \right)^{0} = 3 \cdot 1 = 3 \).

Step 2: Determine the Common Ratio (r)

The common ratio \( r \) is the factor by which each term is multiplied to get the next term. From the series \( 3 \cdot \left( \frac{1}{4} \right)^{k-1} \), it's clear that each subsequent term is multiplied by \( \frac{1}{4} \). Thus, \( r = \frac{1}{4} \).

Step 3: Use the Infinite Series Sum Formula

The sum \( S \) of an infinite geometric series can be calculated using the formula \( S = \frac{a}{1 - r} \) where \( |r| < 1 \). Here, \( a = 3 \) and \( r = \frac{1}{4} \).

Step 4: Calculate the Sum

Substitute \( a = 3 \) and \( r = \frac{1}{4} \) into the formula: \( S = \frac{3}{1 - \frac{1}{4}} = \frac{3}{\frac{3}{4}} \). Simplify this expression by multiplying by the reciprocal: \( S = 3 \cdot \frac{4}{3} = 4 \).

Key Concepts

First Term in Series

In every geometric series, identifying the first term is crucial as it forms the basis for understanding the entire sequence. The first term, often denoted as \(a\), is simply the value of the series when \(k = 1\). This means we substitute \(k = 1\) into the series formula to find this term.
In our given series \(3 \cdot \left(\frac{1}{4}\right)^{k-1}\), we substitute \(k = 1\), resulting in:

• \(a = 3 \cdot \left(\frac{1}{4}\right)^{0} = 3 \cdot 1 = 3\)

Hence, the first term is 3, establishing the starting value of our series.

Common Ratio

The common ratio in a geometric series is the constant factor by which each term is multiplied to obtain the next. It's a key characteristic of a geometric series as it determines the progression of the sequence. Typically, we denote the common ratio by \(r\).
To establish the common ratio of the series \(3 \cdot \left(\frac{1}{4}\right)^{k-1}\), observe that each term is formulated by multiplying the previous term by \(\frac{1}{4}\). This makes:

• \(r = \frac{1}{4}\)

The common ratio being less than 1 and positive ensures that the series converges, allowing us to calculate its infinite sum.

Infinite Series Sum Formula

An infinite geometric series can have a sum when the absolute value of the common ratio \(|r|\) is less than 1. This sum is calculated using a specific formula:
\[ S = \frac{a}{1 - r} \]
where \(a\) is the first term, and \(r\) is the common ratio. Utilizing this formula for our geometric series helps us find the sum effectively.
Given the series with \(a = 3\) and \(r = \frac{1}{4}\), insert these values into the formula:

• \(S = \frac{3}{1 - \frac{1}{4}}\)

Then, simplify the denominator and the equation:

• \(S = \frac{3}{\frac{3}{4}} = 3 \cdot \frac{4}{3} = 4\)

Therefore, the sum of the infinite series converges to 4. This step wisely encapsulates why understanding these elements of a geometric series is vital for finding its sum.