Up until now we've only looked at the sum of the first n terms of a geometric series (S_n). But for some series it is possible to find the sum of an infinite number of terms, and even though that might seem like a lot of work, it's really pretty simple. Let me show you.

From the previous page in this unit, we know that 

S_n = a₁(1 - rⁿ )/(1 - r).

Suppose r was less than 1, but greater than -1. Then what happens to rⁿ when n gets really big?

It gets really, really close to zero, doesn't it?

In fact, we could say that when n goes to infinity, rⁿgoes to zero. And just like that, we have the equation for S, the sum of an infinite geometric series:

S = a₁/(1-r).

It's actually a much simpler equation than the one for the first n terms, but it only works if -1<r<1

Example 1: If the first term of an infinite geometric series is 4, and the common ratio is 1/2, what is the sum?

Solution: S = 4/(1 - 1/2) = 4/(1/2) = 8

Example 2: The sum of an infinite geometric series is 36, and the common ratio is 1/3. What is the first term of the series?

Solution: 363 = a/(1 - 1/3) = a/(2/3), so a = 24.

1.

What is the sum of an infinite geometric series if the first term is 2, and the common ratio is -1/2?

2.

What is the sum of an infinite geometric series if the first term is 1000, and the common ratio is 1/10?

3.

The sum of an infinite geometric series is 20, and the common ratio is 0.2. What is the first term?

4.

The sum of an infinite geometric series is four times the first term. What is the common ratio?

5.

The common ratio in a geometric series is one third the first term. The sum of the infinite series is 0.6. What is the first term?

6.

In series A, the first term is 2, and the common ratio is 0.5. In series B, the first term is 3, and the two infinite series have the same sum. What is the ratio in series B?

7.

In series B, all the terms are twice the corresponding term in series A. If the first term of A is 10, and the ratio of A is 0.2, what is the sum of series B?

8.

Find the sum of an infinite geometric series if the first term and the common ratio are both -1/3.

9.

The sum of an infinite geometric series is 24, and the sum of the first 200 terms of the series is also 24. What are the first term and common ratio of the series?

10.

Explain why our formula only works if r is between -1 and 1.