A geometric sequence is infinite if it continues indefinitely without a last term. This is usually implied by the problem statement or how the sequence is presented.
Here's how you can identify an infinite geometric sequence:
- Explicit Indication: The sequence is explicitly stated to be infinite. For example, "Consider the infinite geometric sequence..."
- Ellipsis (...) at the End: The sequence is written with an ellipsis at the end, indicating that it continues without end. For example: 2, 4, 8, 16, ...
- Formula for the nth Term: You are given a formula for the nth term, but n is not restricted to a finite set of numbers. This means n can be any positive integer.
- Implied Context: The problem's context implies infinity. For example, problems related to convergence or divergence of series usually deal with infinite sequences.
- No Defined Last Term: The sequence is described in a way that doesn't define an end or a last term.
Example:
The sequence 1, 1/2, 1/4, 1/8, ... is an infinite geometric sequence because it has an ellipsis at the end. The common ratio is 1/2, and the terms continue to get smaller, approaching zero, but never actually reaching it.
Contrast with a finite geometric sequence:
A finite geometric sequence has a defined last term. For example: 1, 2, 4, 8, 16. This sequence has five terms and ends with 16.
In summary, look for explicit statements, ellipses, unrestricted formulas for the nth term, or contextual clues to determine if a geometric sequence is infinite.
