The formula for the sum of an infinite geometric series is: s = a₁ / (1 - r), where s is the sum, a₁ is the first term, and r is the common ratio, provided that |r| < 1.
To understand this formula fully, let's break it down:
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Infinite Geometric Series: This is a series where each term is multiplied by a constant value (the common ratio) to obtain the next term, and the series continues infinitely.
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a₁ (First Term): This is the initial value of the sequence.
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r (Common Ratio): This is the constant value that is multiplied by each term to get the next term. You can find r by dividing any term by its preceding term (e.g., a₂ / a₁). Crucially, for the sum of an infinite geometric series to exist (i.e., converge to a finite value), the absolute value of the common ratio must be less than 1 (|r| < 1). If |r| ≥ 1, the series diverges and does not have a finite sum.
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s (Sum): This represents the finite value that the infinite series approaches as more and more terms are added, only if |r| < 1.
When Does the Formula Apply?
The formula s = a₁ / (1 - r) only applies when the absolute value of the common ratio r is less than 1 (|r| < 1). This condition is essential for the series to converge to a finite sum. If |r| ≥ 1, the series diverges, meaning the sum grows infinitely large (or oscillates) and doesn't have a defined value.
Example:
Consider the infinite geometric series: 1 + 1/2 + 1/4 + 1/8 + ...
- a₁ = 1 (the first term)
- r = 1/2 (each term is half of the previous term)
Since |1/2| < 1, we can use the formula:
s = 1 / (1 - 1/2) = 1 / (1/2) = 2
Therefore, the sum of this infinite geometric series is 2.
In summary, the formula s = a₁ / (1 - r) calculates the sum of an infinite geometric series only when the absolute value of the common ratio r is less than 1. Otherwise, the series diverges and does not have a finite sum.
