Understanding Geometric Series Convergence

The question asks us to determine the conditions under which the given geometric series, $ 1 + r + r^2 + r^3 + \dots $, converges or diverges. We are given the constraint that the common ratio $ r $ must be greater than 0 (i.e., $ r > 0 $).

A geometric series is characterized by a first term and a common ratio ($ r $). Each term in the series is obtained by multiplying the previous term by this common ratio. The general form is $ a + ar + ar^2 + ar^3 + \dots $.

Standard Convergence Rule for Geometric Series

The behavior (convergence or divergence) of an infinite geometric series depends critically on the value of its common ratio, $ r $. The standard mathematical rule is:

• Convergence: The series converges if the absolute value of the common ratio is strictly less than 1. Mathematically, this is expressed as $ |r| < 1 $.
• Divergence: The series diverges if the absolute value of the common ratio is greater than or equal to 1. Mathematically, this is expressed as $ |r| \ge 1 $.

Applying the Rule with the Given Condition ($ r > 0 $)

For the specific series $ 1 + r + r^2 + r^3 + \dots $, the first term $ a = 1 $ and the common ratio is $ r $. We are given that $ r > 0 $. Let's apply the standard convergence rule under this condition:

Convergence Case: $ 0 < r < 1 $

When the common ratio $ r $ is between 0 and 1 (i.e., $ 0 < r < 1 $), it satisfies the condition $ |r| < 1 $ because $ r $ is positive. Therefore, in this interval, the geometric series converges.

Divergence Case: $ r \ge 1 $

When the common ratio $ r $ is greater than or equal to 1 (i.e., $ r \ge 1 $), it satisfies the condition $ |r| \ge 1 $ because $ r $ is positive. Therefore, in this interval, the geometric series diverges.

Evaluating the Options Provided

Let's examine each option in light of our analysis:

Option 1 Analysis

Statement: "It diverges, if $ 0 < r < 1 $ and converges, if $ r\ge 1 $". This statement incorrectly assigns the conditions. Our analysis shows the opposite is true.

Option 2 Analysis

Statement: "It converges, if $ 0 < r < 1 $ and diverges, if $ r\ge 1 $". This statement accurately reflects the convergence and divergence conditions for the given geometric series with $ r > 0 $. This aligns with our derived results.

Option 3 Analysis

Statement: "It is always convergent". This is false because the series diverges when $ r \ge 1 $.

Option 4 Analysis

Statement: "It is always divergent". This is false because the series converges when $ 0 < r < 1 $.

Option 5 Analysis

This option is empty and does not present a valid statement.

Conclusion

The geometric series $ 1 + r + r^2 + r^3 + \dots $ converges when the common ratio $ r $ is between 0 and 1 ($ 0 < r < 1 $) and diverges when the common ratio $ r $ is 1 or greater ($ r \ge 1 $).