The sum of an infinite Geometric Progression (GP) where the common ratio 'r' is greater than 1 is undefined and tends to infinity (±∞).
Understanding Geometric Progressions and their Sum
A Geometric Progression (GP), also known as a geometric sequence, is a sequence of numbers where each term is found by multiplying the previous term by a constant factor called the common ratio ('r').
The general form of a GP is: a, ar, ar², ar³, ... where 'a' is the first term.
The sum of the first 'n' terms of a GP is given by:
Sn = a(1 - rn) / (1 - r), if r ≠ 1
When dealing with infinite geometric progressions, the behavior of the sum depends critically on the value of the common ratio 'r'.
Infinite GP and the Common Ratio
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|r| < 1: If the absolute value of 'r' is less than 1, as 'n' approaches infinity, rn approaches 0. Therefore, the sum of the infinite GP converges to a finite value: S∞ = a / (1 - r).
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|r| ≥ 1: If the absolute value of 'r' is greater than or equal to 1, as 'n' approaches infinity, rn does not approach 0. Instead, it either approaches infinity (if r > 1) or oscillates (if r ≤ -1 or r = 1). In these cases, the sum of the infinite GP diverges, meaning it does not approach a finite limit and is considered undefined. For r > 1, the sum grows without bound, tending to positive or negative infinity depending on the sign of 'a'. If r=1, the sum becomes infinity. If r is less than -1, the sum oscillates and does not converge.
Why the Sum is Undefined for r > 1
When r > 1, each subsequent term in the GP is larger than the previous one. As you add more and more of these increasingly large terms, the sum grows without any limit. Therefore, the sum approaches infinity. The expression a / (1 - r) is not valid for r >= 1.
Example
Consider the GP: 2, 4, 8, 16, ...
Here, a = 2 and r = 2 (which is greater than 1).
As you keep adding terms:
2 + 4 = 6
2 + 4 + 8 = 14
2 + 4 + 8 + 16 = 30
2 + 4 + 8 + 16 + 32 = 62
The sum keeps increasing rapidly, tending towards infinity.
Conclusion
Therefore, the sum of an infinite geometric progression where the absolute value of the common ratio 'r' is greater than or equal to 1 is undefined and typically considered to be infinity (±∞).
