Problem 94 - Algebra and Trigonometry

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 or converges if the absolute value of the common ratio 'r' is less than 1. The sum 'S' of the series can be calculated using the formula S = a / (1 - r), where 'a' is the first term of the series. If the absolute value of the common ratio 'r' is greater than or equal to 1, the series diverges and does not have a sum.

Step 1: Identify a and r

First, identify the first term 'a' and the common ratio 'r' of the series. The first term 'a' is the first value in the series. The common ratio 'r' is found by dividing the second term by the first term.

Step 2: Check for converge or diverge

Next, determine whether the series converges or diverges. If the absolute value of 'r' is less than 1, then the series converges. But, if the absolute value of 'r' is greater than or equal to 1, then the series diverges.

Step 3: Calculate the sum of the series

If the series converges, calculate the sum of the series using the formula S = a / (1 - r). It's important to note that the sum is only valid for a converging series.