How Do You Find the Ratio of an Infinite Geometric Series?

To find the common ratio (r) of an infinite geometric series, you need to examine the relationship between consecutive terms. The common ratio is the constant value by which each term is multiplied to obtain the next term.

Methods for Finding the Common Ratio (r):

1. Using Two Consecutive Terms: The simplest method involves dividing any term by the preceding term. If a_n represents the nth term, then the common ratio is:

   r = a_n+1 / a_n

   This works for any pair of consecutive terms in the series. For example:

   • Given the series 2, 4, 8, 16..., r = 4/2 = 8/4 = 16/8 = 2

2. Using the Sum and First Term (for Convergent Series): For an infinite geometric series to have a finite sum, the absolute value of the common ratio must be less than 1 (|r| < 1). If you know the sum (S) and the first term (a₁), you can use the formula for the sum of an infinite geometric series:

   S = a₁ / (1 - r)

   Solving for r, we get:

   r = 1 - (a₁ / S)

   Example: If S = 2 and a₁ = 0.5, then r = 1 - (0.5 / 2) = 0.75

Important Considerations:

• Convergence: An infinite geometric series only converges (has a finite sum) if |r| < 1. If |r| ≥ 1, the series diverges and does not have a finite sum.
• Series Definition: You must have a clearly defined geometric series. Each term must be obtained by multiplying the previous term by a constant value (the common ratio).

Example using the formula a2⁄a1:

Let's consider the series: 3, 6, 12, 24...

• a₁ (first term) = 3
• a₂ (second term) = 6

The common ratio (r) is: r = a₂ / a₁ = 6 / 3 = 2

Therefore, the common ratio of this geometric series is 2. Note that since |r| (2) is not less than 1, this series diverges and does not have a finite sum.