The sum of an infinite arithmetic progression (AP) is not a finite value and, depending on the common difference, either diverges to positive or negative infinity or is undefined.
Understanding Infinite Arithmetic Progressions
An arithmetic progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference, denoted as 'd'. An infinite AP continues indefinitely. The behavior of the sum of its terms depends on the common difference:
Infinite AP Formula Analysis
According to the provided reference, the sum of an infinite arithmetic progression behaves as follows:
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When d < 0: The sum of the infinite AP is −∞ (negative infinity). This occurs because the terms keep getting smaller and smaller (more negative). When you keep adding such a sequence of numbers, the sum will keep decreasing without limit.
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When d > 0: The sum of an infinite AP is +∞ (positive infinity). This happens when each term is larger than the previous one and the sum grows unboundedly.
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When d = 0: The sequence becomes a constant sequence, for example, 5, 5, 5, ... The sum of such a sequence would go to positive or negative infinity depending on the number, but the question only talks about sequences that aren't constant so we can exclude this.
Key Takeaways
- There is no single formula to give a finite sum for an infinite AP, unlike finite APs.
- The behavior of the sum is critically dependent on the common difference (d).
- If d is not zero, the sum will diverge to either positive or negative infinity depending on the sign of d.
Practical Implications
Since infinite APs generally don't have a finite sum, the concept is more theoretical than practically applicable. In real-world scenarios, arithmetic progressions are usually finite.
| Common Difference (d) | Sum of Infinite AP |
|---|---|
| d < 0 | -∞ |
| d > 0 | +∞ |
