What characterizes a divergent geometric series?

What characterizes a divergent geometric series?

• A. Its terms converge to zero
• B. Its sum tends towards infinity
• C. It contains only positive terms
• D. Its terms are all negative

Answer: (B)

A divergent geometric series is characterized by the fact that as more terms are added, the sum grows without bound, typically tending towards infinity. This phenomenon occurs when the common ratio of the series is greater than or equal to one in absolute value. For instance, if the common ratio is greater than one, each successive term increases in magnitude, leading the total sum to grow indefinitely. In more detail, a geometric series can be expressed as \( a + ar + ar^2 + ar^3 + \ldots \), where \( a \) represents the first term and \( r \) is the common ratio. When \( |r| \geq 1 \), the series does not converge to a finite value; instead, the sum increases without limitation. Thus, the distinguishing feature of a divergent geometric series is indeed that its sum tends towards infinity. Regarding the other options, while a divergent series can contain positive or negative terms, it is the nature of the sum that primarily defines its divergence. Therefore, the focus here is specifically on the behavior of the sum itself, affirming that the correct characterization of a divergent geometric series can be accurately described as having its sum tending towards infinity.

A divergent geometric series is characterized by the fact that as more terms are added, the sum grows without bound, typically tending towards infinity. This phenomenon occurs when the common ratio of the series is greater than or equal to one in absolute value. For instance, if the common ratio is greater than one, each successive term increases in magnitude, leading the total sum to grow indefinitely.

In more detail, a geometric series can be expressed as ( a + ar + ar^2 + ar^3 + \ldots ), where ( a ) represents the first term and ( r ) is the common ratio. When ( |r| \geq 1 ), the series does not converge to a finite value; instead, the sum increases without limitation. Thus, the distinguishing feature of a divergent geometric series is indeed that its sum tends towards infinity.

Regarding the other options, while a divergent series can contain positive or negative terms, it is the nature of the sum that primarily defines its divergence. Therefore, the focus here is specifically on the behavior of the sum itself, affirming that the correct characterization of a divergent geometric series can be accurately described as having its sum tending towards infinity.