To add an infinite series, we typically use specific formulas or techniques, particularly if the series is a geometric series. A geometric series is a series where each term after the first is found by multiplying the previous term by a constant value (the common ratio). If the common ratio is between -1 and 1, then we can determine the sum of the infinite series using a specific formula.
Understanding Geometric Series
A geometric series can be represented as:
a + ar + ar² + ar³ + ar⁴ + ...
Where:
- a is the first term.
- r is the common ratio.
Formula for Summing Infinite Geometric Series
The formula for the sum (S) of an infinite geometric series when -1 < r < 1 is given by:
S = a / (1 - r)
This formula is valid only when the absolute value of the common ratio, denoted as |r|, is less than 1. If |r| ≥ 1, the sum of the series is infinite, meaning it doesn't converge to a single number.
Steps to Sum an Infinite Geometric Series:
- Identify the First Term (a): Determine the value of the first term in the series.
- Determine the Common Ratio (r): Divide any term by the term that comes before it. This value should be constant throughout the series.
- Check the Convergence Condition: Make sure that -1 < r < 1 (or |r| < 1). If this condition is not met, the series diverges, and the sum does not converge to a finite number.
- Apply the Sum Formula: If the condition is met, use the formula
S = a / (1 - r)to find the sum.
Example:
Let's say we have the infinite geometric series:
2 + 1 + 1/2 + 1/4 + ...
Here:
- a = 2 (the first term)
- r = 1/2 (the common ratio, since 1/2 = 1/2, (1/2)/1=1/2, (1/4) / (1/2) = 1/2)
Since -1 < 1/2 < 1, we can use the formula:
S = 2 / (1 - 1/2) = 2 / (1/2) = 2 * 2 = 4
Therefore, the sum of this infinite geometric series is 4.
Key Points
- The formula
S = a / (1 - r)is only applicable for convergent geometric series, where -1 < r < 1. - Not all infinite series have a sum.
- For other types of infinite series, more advanced techniques may be required to determine their sum, if it exists.
- If |r| ≥ 1, the geometric series does not converge to a finite sum.
