The sum of an infinite arithmetic series is either positive infinity (+∞) or negative infinity (-∞), unless the common difference is zero, in which case the sum will be infinitely large or small depending on the first term.
Understanding Arithmetic Series
An arithmetic series is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the "common difference" (d). The general form of an arithmetic series is:
a, a + d, a + 2d, a + 3d, ...
Where 'a' is the first term.
Why Infinite Arithmetic Series Diverge
Unlike geometric series, arithmetic series generally do not converge to a finite value when summed to infinity. This is because:
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Terms don't approach zero: In a typical arithmetic series (where d is not zero), the terms keep increasing (if d > 0) or decreasing (if d < 0) without bound. They never approach zero, a necessary (but not sufficient) condition for convergence.
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The sum keeps growing: As you add more and more terms, the sum will continue to increase (towards +∞) if d is positive or decrease (towards -∞) if d is negative.
Cases and Examples
Let's illustrate with examples:
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Example 1: Positive Common Difference (d > 0)
Consider the series: 1 + 2 + 3 + 4 + ...
Here, a = 1 and d = 1. The sum of this infinite series approaches positive infinity (+∞).
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Example 2: Negative Common Difference (d < 0)
Consider the series: 5 + 4 + 3 + 2 + 1 + 0 + (-1) + (-2) + ...
Here, a = 5 and d = -1. The sum of this infinite series approaches negative infinity (-∞).
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Example 3: Zero Common Difference (d = 0)
Consider the series: 2 + 2 + 2 + 2 + ...
Here, a = 2 and d = 0. The sum of this infinite series approaches positive infinity (+∞). The sum is unbounded and thus diverges.
Comparison with Geometric Series
It's crucial to distinguish this from geometric series. A geometric series has a common ratio (r) between consecutive terms, not a common difference. Infinite geometric series can converge if the absolute value of the common ratio is less than 1 (|r| < 1). In such cases, the sum converges to a finite value given by the formula:
S = a / (1 - r)
Where 'a' is the first term and 'r' is the common ratio. This convergence does not occur with arithmetic series.
Summary
In summary, the sum of an infinite arithmetic series typically diverges to either positive or negative infinity because the terms do not approach zero, and the sum continues to increase or decrease without bound. Only in the trivial case where the common difference is zero does the series diverge to infinity (positive or negative) or equal to zero.
