Problem 94 - College Algebra

Question

How do you determine if an infinite geometric series has a sum? Explain how to find the sum of such an infinite geometric series.

Step-by-Step Solution

Answer

An infinite geometric series has a sum if the absolute value of the common ratio is less than 1, otherwise it diverges. The sum of such a series can be found using the formula \(S = a / (1 - r)\), where 'a' is the first term of the series, and 'r' is the common ratio.

Step 1: Identify if the series is a geometric series

To determine if a series is a geometric series, confirm if each term after the first is a constant multiple of the previous term. If so, the series is a geometric series and the constant is the common ratio.

Step 2: Determine the common ratio

Calculate the common ratio (r) by dividing a term by its preceding term. Do this for multiple terms to ensure the ratio is the same each time.

Step 3: Check the absolute value of the common ratio

To know if the infinite geometric series has a sum, check if the absolute value of the common ratio (|r|) is less than 1. If |r| < 1, then the infinite geometric series converges and has a sum. If |r| >= 1, then the series diverges and does not have a finite sum.

Step 4: Calculate the sum of the series

If |r| < 1, use the formula for finding the sum of an infinite geometric series: \(S = a / (1 - r)\), where 'S' is the sum of the series, 'a' is the first term in the series, and 'r' is the common ratio. Substitute the known values of 'a' and 'r' into this formula to calculate 'S'.