Problem 42 - College Algebra

Question

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

Step-by-Step Solution

Answer

The sum of this infinite geometric series is 9/4 or 2.25

Step 1: Identify the First Term

The first term (a) of the geometric series is 3, therefore, \( a = 3 \)

Step 2: Identify the Common Ratio

To find the common ratio (r), divide any term by its preceding term. Let's take the second term (-1) and divide by the first term (3). So, \( r = -1/3 \)

Step 3: Apply the Sum Formula

Use the formula for the sum of an infinite geometric series. This is \( S = a / (1 - r) \). Substituting the values of \( a \) and \( r \) gives \( S = 3 / (1 - (-1/3)) \)

Step 4: Simplify the Expression

Simplify the expression by performing the operations. First, calculate the denominator, 1 - (-1/3) equals 4/3. So the expression becomes \( S = 3 / (4/3) \)

Step 5: Final Calculation

The fraction in the denominator signifies division. Hence, the expression becomes \( S = 3 ÷ 4/3 = 3 × 3/4 = 9/4 \)