What do you know about the common ratio of an infinite geometric series whose sum is finite?

For an infinite geometric series to have a finite sum, the common ratio must meet a specific condition: its absolute value must be less than 1.

Understanding the Common Ratio and Finite Sums

An infinite geometric series is a sum of the form:

a + ar + ar² + ar³ + ...

Where:

• a is the first term.
• r is the common ratio.

Whether this sum converges to a finite value depends entirely on the value of r.

The Convergence Condition

The key condition for an infinite geometric series to have a finite sum is:

|r| < 1

In other words, the absolute value of the common ratio r must be between 0 and 1 ( -1 < r < 1).

• If |r| < 1: The series converges, meaning its sum approaches a finite value. The formula for the sum (S) is:

  S = a / (1 - r)

  (As provided in the reference.)

• If |r| ≥ 1: The series diverges, meaning its sum grows without bound (approaches infinity or negative infinity).

Examples

Here are a few examples to illustrate the concept:

• Convergent Series (Finite Sum):

  Consider the series: 1 + 1/2 + 1/4 + 1/8 + ...

  • a = 1
  • r = 1/2
  • Since |1/2| < 1, the series converges.
  • The sum is: S = 1 / (1 - 1/2) = 1 / (1/2) = 2

• Divergent Series (Infinite Sum):

  Consider the series: 1 + 2 + 4 + 8 + ...

  • a = 1
  • r = 2
  • Since |2| > 1, the series diverges. The sum goes to infinity.

• Divergent Series (Oscillating):

  Consider the series: 1 - 1 + 1 - 1 + ...

  • a = 1
  • r = -1
  • Since |-1| = 1, the series diverges. The sum oscillates between 0 and 1.

Summary Table

Condition	Convergence/Divergence	Sum
	r	< 1
	r	≥ 1

In conclusion, the absolute value of the common ratio r of an infinite geometric series must be strictly less than 1 for the series to have a finite sum.