The sum of an infinite arithmetic progression is either positive or negative infinity, depending on the common difference.
Understanding Infinite Arithmetic Progressions
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, often denoted by 'd'. An infinite arithmetic progression continues indefinitely.
Determining the Sum
Whether an infinite arithmetic progression converges (has a finite sum) or diverges (approaches infinity) depends entirely on the common difference, 'd'.
Divergence to Infinity (∞)
According to our reference, the sum of an infinite arithmetic progression is ∞ if the common difference is greater than 0. This occurs because each subsequent term is larger than the previous one, leading to an ever-increasing sum.
- Example: The arithmetic progression 1, 2, 3, 4, ... has a common difference of 1 (d = 1). As you continue adding terms, the sum grows infinitely large.
Divergence to Negative Infinity (-∞)
Conversely, the sum of an infinite arithmetic progression is -∞ if the common difference is less than 0. In this scenario, each subsequent term is smaller than the previous one, leading to an ever-decreasing (more negative) sum.
- Example: The arithmetic progression -1, -2, -3, -4, ... has a common difference of -1 (d = -1). As you continue adding terms, the sum grows infinitely negative.
Table Summary
| Common Difference (d) | Sum of Infinite Arithmetic Progression |
|---|---|
| d > 0 | ∞ |
| d < 0 | -∞ |
Key Takeaway
An infinite arithmetic progression never converges to a finite sum. It always diverges to either positive or negative infinity, determined solely by the sign of the common difference.
