The absolute value of 'r' determines convergence/divergence: if 0 < |r| < 1, the series converges; if |r| ≥ 1, the series diverges.

Changing the starting index (e.g., from n=0 to n=1) does not affect whether the series converges or diverges.

The geometric series test provides a straightforward condition based on the common ratio 'r' to determine if an infinite geometric series converges or diverges.

$$\sum_{n=0}^{\infty} a cdot r^n$$

$$\sum_{n=1}^{\infty} a cdot r^{n-1}$$

$$\sum_{n=1}^{\infty} acdot r^{n-1} = \sum_{n=0}^{\infty} a cdot r^n = \frac{a}{1-r}$$

A series with a constant ratio between successive terms.

'a' is the initial term of the geometric series.

'r' is the common ratio between consecutive terms in a geometric series.

A series converges if its partial sums approach a finite limit.

A series diverges if its partial sums do not approach a finite limit.