Question

What are the different conditions of a that result in different results of convergence/divergence for the series ∑ n=0 to infinity a^n?

A. |a| < 1.
B. |a| > 1.
C. |a| = 1.
D. All of the above.

Answer 1

The different conditions that result in different results of convergence/divergence for the series ∑ n=0 to infinity a^n are |a| < 1 (converges), |a| > 1 (diverges), and |a| = 1 (may converge or diverge).

Step-by-step explanation:

The different conditions of a that result in different results of convergence/divergence for the series ∑ n=0∞ an are:

1. |a| < 1: When the absolute value of a is less than 1, the series converges. For example, if a is 0.5, the series would converge to a finite value.
2. |a| > 1: When the absolute value of a is greater than 1, the series diverges. For example, if a is 2, the series would diverge and not have a finite sum.
3. |a| = 1: When the absolute value of a is equal to 1, the series may converge or diverge depending on the specific values of a. This condition requires further analysis.

So, the correct answer is D. All of the above.