In a geometric series, if the common ratio is greater than 1, what can be said about the series?

In a geometric series, the series is defined by the formula:

[

S = a + ar + ar^2 + ar^3 + \ldots

]

where (a) is the first term and (r) is the common ratio. The behavior of a geometric series greatly depends on the value of the common ratio (r).

When the common ratio is greater than 1, this means each subsequent term in the series is larger than the previous one. Specifically, as the terms continue to be added, the series grows without bound. Mathematically, this can be observed as each added term significantly increases the total sum, leading to the series diverging to infinity. In this case, as (n) approaches infinity:

[

S_n = a(1 + r + r^2 + \ldots + r^{n-1}) \to \infty \text{ when } r > 1.

]

In contrast, a series that converges to a finite limit occurs when the common ratio is between -1 and 1 (exclusive), which allows the terms to decrease in magnitude. If the series oscillates, it would not follow a consistent pattern of growth, and a steady