Question

1. which statement about divergent infinite geometric series is true?

• the graph of a divergent infinite geometric series curves toward its sum.
• the graph of a divergent infinite geometric series curves toward infinity.
• all infinite geometric series have divergent graphs.
• a divergent infinite geometric series will have a common ratio of 0.50.

Explanation:

🆕 New Concept Discovered: Infinite Geometric Series Convergence and Divergence
Whether a sum approaches a number or grows without bound

Step 1: Define Divergence in Infinite Series

An infinite geometric series has the form:

$$ S = a_1 + a_1r + a_1r^2 + a_1r^3 + \dots $$

• Convergent series: If the absolute value of the common ratio is less than 1 (\(|r| < 1\)), the terms get smaller and smaller, approaching 0. The sum of the series approaches a specific, finite number.
• Divergent series: If the absolute value of the common ratio is greater than or equal to 1 (\(|r| \ge 1\)), the terms do not approach 0. As you add more terms, the partial sums do not settle on a single number; instead, they grow infinitely larger (toward positive or negative infinity) or oscillate.

Step 2: Evaluate the Given Statements

• "The graph of a divergent infinite geometric series curves toward its sum."
• False. A divergent series does not have a finite sum to curve toward.
• "The graph of a divergent infinite geometric series curves toward infinity."
• True. Because the terms do not shrink to 0, the cumulative sum grows without bound, meaning its graph curves toward infinity.
• "All infinite geometric series have divergent graphs."
• False. If \(|r| < 1\), the series converges to a finite sum.
• "A divergent infinite geometric series will have a common ratio of 0.50."
• False. A common ratio of \(r = 0.50\) satisfies \(|r| < 1\), which means the series would be convergent, not divergent.

Answer:

The graph of a divergent infinite geometric series curves toward infinity.