An infinite geometric series differs from a finite geometric series primarily in whether it has a limited number of terms, and crucially, in whether it converges to a finite sum.
Here's a breakdown of the key differences:
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Number of Terms: A finite geometric series has a specific, countable number of terms (e.g., 5 terms, 100 terms). An infinite geometric series has an unlimited number of terms, extending indefinitely.
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Summation: You can always calculate the sum of a finite geometric series using the formula:
Sn = a(1 - rn) / (1 - r), where:
- Sn is the sum of the first n terms.
- a is the first term.
- r is the common ratio.
- n is the number of terms.
The sum of an infinite geometric series, however, exists only when the absolute value of the common ratio r is less than 1 (|r| < 1). In this case, the series is said to converge. The formula for the sum of a convergent infinite geometric series is:
S = a / (1 - r), where:
- S is the sum to infinity.
- a is the first term.
- r is the common ratio (|r| < 1).
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Convergence vs. Divergence: This is the most significant difference. A finite geometric series always has a finite sum. An infinite geometric series may have a finite sum (if |r| < 1), in which case it's said to converge. If |r| ≥ 1, the infinite geometric series diverges, meaning the sum increases without bound and does not approach a finite value.
Here's a table summarizing the differences:
| Feature | Finite Geometric Series | Infinite Geometric Series |
|---|---|---|
| Number of Terms | Limited, countable | Unlimited |
| Sum | Always has a finite sum | Has a finite sum only if |
| Common Ratio (r) | No restrictions on the value of r. |
Example:
- Finite: 2 + 4 + 8 + 16 (a = 2, r = 2, n = 4). The sum can be calculated.
- Infinite (Convergent): 1 + 1/2 + 1/4 + 1/8 + ... (a = 1, r = 1/2). The sum converges to 2.
- Infinite (Divergent): 1 + 2 + 4 + 8 + ... (a = 1, r = 2). The sum diverges to infinity.
In essence, an infinite geometric series introduces the crucial concept of convergence, which determines whether the sum of an infinite number of terms approaches a finite value. This convergence depends entirely on the common ratio. A finite geometric series, on the other hand, always has a calculable, finite sum.
