The formula for the sum to infinity of a geometric series is S_∞ = a / (1 - r).
This powerful mathematical tool allows us to calculate the total sum of an infinitely long sequence of numbers, provided certain conditions are met. It's particularly useful in various fields, from finance to physics, for understanding patterns that extend indefinitely.
Understanding the Components of the Formula
To effectively use the sum to infinity formula, it's essential to understand its key components:
| Component | Description |
|---|---|
| S_∞ | Represents the sum to infinity of the geometric series. This is the value the sum of the terms approaches as the number of terms increases without bound. |
| a | Denotes the first term of the geometric series. This is the very first number in your sequence. |
| r | Stands for the common ratio. This is the constant number by which each term is multiplied to get the next term in the sequence. To determine the common ratio, you can divide any term in the series by its immediately preceding term. For example, if you have a series x, y, z..., then r = y/x or r = z/y. |
Condition for Convergence
A crucial aspect of the sum to infinity formula is that it only applies when the series converges. A geometric series converges if and only if the absolute value of its common ratio (r) is less than 1.
Mathematically, this condition is expressed as:
|r| < 1
This means that r must be between -1 and 1 (i.e., -1 < r < 1). If |r| ≥ 1, the terms of the series will either stay the same size or grow larger, meaning the sum will not approach a finite value, and thus, the sum to infinity does not exist (or diverges).
How to Calculate the Sum to Infinity
To calculate the sum to infinity, follow these simple steps:
- Identify the First Term (a): Look at the given series and note the initial value.
- Calculate the Common Ratio (r): Divide the second term by the first term (or any term by its preceding term) to find
r. - Check the Convergence Condition: Ensure that
|r| < 1. If it's not, the sum to infinity does not exist. - Apply the Formula: Substitute the values of
aandrinto the formulaS_∞ = a / (1 - r)and compute the result.
Practical Example
Let's find the sum to infinity for the geometric series: 5 + 2.5 + 1.25 + 0.625 + ...
-
Identify the first term (a):
The first term,a, is 5. -
Calculate the common ratio (r):
Divide the second term by the first term:r = 2.5 / 5 = 0.5.
(You could also check1.25 / 2.5 = 0.5, confirming the common ratio). -
Check the convergence condition:
The common ratior = 0.5. Since|0.5| < 1, the series converges, and a sum to infinity exists. -
Apply the formula:
S_∞ = a / (1 - r)
S_∞ = 5 / (1 - 0.5)
S_∞ = 5 / 0.5
S_∞ = 10
Therefore, the sum to infinity of the series 5 + 2.5 + 1.25 + 0.625 + ... is 10. This means that as you add more and more terms of this sequence, their total sum gets closer and closer to 10.
