The sum of an infinite geometric series with a common ratio whose absolute value is less than 1 (i.e., -1 < r < 1) is given by the formula: S = a / (1 - r), where 'a' is the first term of the series, and 'r' is the common ratio.
Understanding the Formula
Geometric Series Basics
A geometric series is a sequence where each term is multiplied by a constant value called the common ratio ('r') to obtain the next term. A general form of a geometric series is:
a + ar + ar2 + ar3 + ...
where:
- 'a' is the first term
- 'r' is the common ratio
Why the Ratio Must Be Less Than 1
The formula S = a / (1 - r) only works when |r| < 1. When the absolute value of the common ratio is greater than or equal to 1, the series diverges; meaning the sum approaches infinity or negative infinity (and therefore doesn't have a finite sum). Intuitively, if each term is multiplied by a number greater than or equal to 1, the terms will not approach zero, and you'll be adding larger and larger values indefinitely.
Derivation of the Formula
While not strictly necessary to answer the question, understanding the derivation can provide greater insight:
- Let S = a + ar + ar2 + ar3 + ...
- Multiply both sides by 'r': rS = ar + ar2 + ar3 + ar4 + ...
- Subtract the second equation from the first: S - rS = a
- Factor out S: S(1 - r) = a
- Solve for S: S = a / (1 - r)
This derivation is valid only if the series converges, which requires |r| < 1.
Example
Let's say we have a geometric series: 2 + 1 + 0.5 + 0.25 + ...
Here, a = 2 (the first term) and r = 0.5 (each term is half of the previous term).
Using the formula:
S = a / (1 - r) = 2 / (1 - 0.5) = 2 / 0.5 = 4
Therefore, the sum of this infinite geometric series is 4.
Summary
In conclusion, the sum of an infinite geometric series with a common ratio 'r' where |r| < 1 is found using the formula S = a / (1 - r), where 'a' is the first term.
