The sum of a geometric series with a common ratio (r) greater than 1 does not exist.
Explanation
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). The sum of an infinite geometric series can be calculated only under specific conditions, as highlighted in the reference provided:
The infinite sum of a geometric sequence can be calculated if the common ratio is a fraction between −1 and 1 (that is |r|<1) as follows: S∞=a11−r. If |r|≥1, then no sum exists.
This means that if the absolute value of the common ratio (|r|) is greater than or equal to 1, the series diverges, and its sum approaches infinity, therefore having no finite sum.
Why No Sum Exists When r ≥ 1
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Divergence: When r is greater than 1, each subsequent term in the series becomes larger than the previous one. This causes the sum to grow infinitely large. For example, consider the series 2 + 4 + 8 + 16 +... where r=2. This sum will grow indefinitely and not converge to a finite number.
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|r| = 1 Cases:
- If r = 1, the series is just a constant value added repeatedly (e.g., 2+2+2+...). The sum will also grow infinitely large.
- If r = -1, the series oscillates between two values without approaching a single sum (e.g. 2 - 2 + 2 -2 + ...), and it does not have a sum.
Summary
| Condition | Does a Sum Exist? |
|---|---|
| r | |
| r |
In conclusion, for a geometric series where the common ratio 'r' is greater than 1, there is no finite sum. The series diverges.
