A geometric series is the sum of the terms of a geometric sequence. Each term in the sequence is multiplied by a constant, called the common ratio, to get the next term.
An infinite geometric series is a geometric series with an infinite number of terms. It continues without end.
A finite geometric series is a geometric series with a limited number of terms. It has a definite start and end.
| Feature | Infinite Geometric Series | Finite Geometric Series |
|---|---|---|
| Number of Terms | Infinite | Finite |
| Sum | May converge to a finite value (if |r| < 1) or diverge. | Always has a finite sum. |
| Common Ratio (r) | Convergence depends on the value of 'r'. If $|r| < 1$, the series converges. | The value of 'r' does not affect the finiteness of the sum. |
| Formula for Sum | If $|r| < 1$: $S = \frac{a}{1-r}$, otherwise, the sum diverges. | $S_n = \frac{a(1-r^n)}{1-r}$, where 'n' is the number of terms. |
| Examples | $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$ | $1 + 2 + 4 + 8 + 16$ |
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