To solve this problem, let's analyze the information step by step:

Given:

1. The power series $\sum_{n=0}^\infty c_n x^n$ converges when $x = 2$.
2. The series diverges when $x = -7$.

Conceptual Analysis:

• The radius of convergence $R$ of a power series $\sum c_n x^n$ determines the interval in which the series converges absolutely.
• The series converges for $|x| < R$ and diverges for $|x| > R$.
• Knowing that it converges at $x = 2$, we conclude that $R \geq 2$ because $x = 2$ is within the radius of convergence.
• Knowing that it diverges at $x = -7$, we conclude that $R < 7$, because $x = -7$ lies outside the radius of convergence.

Key Insight:

From the above, the radius of convergence $R$ satisfies $2 \leq R < 7$.

Answer Evaluation:

Let's check the options:

1. (A) The radius of convergence is $R = 2$:

   • This is not necessarily true, as $R$ could be larger than 2 (but smaller than 7).

2. (B) The power series diverges at $x = 8$:

   • True, because $x = 8$ lies outside any possible radius of convergence ($R < 7$).

3. (C) The power series converges at $x = -2$:

   • We cannot determine this, as $x = -2$ may or may not lie within the radius of convergence (depends on the exact value of $R$).

4. (D) The power series converges at $x = -4$:

   • Same as (C), it is indeterminate.

5. (E) The power series diverges at $x = 7$:

   • True, since $x = 7$ lies outside the radius of convergence ($R < 7$).

Correct Answer:

• Both (B) and (E) are true, but the most direct conclusion is (E): The power series diverges at $x = 7$.

Let me know if you'd like further clarifications or more detailed reasoning!

Related Questions:

1. What is the definition of a power series' radius of convergence?
2. How do endpoints of the interval of convergence behave in such problems?
3. What happens if a power series diverges at a point closer than the radius $R$?
4. How is absolute convergence related to the radius of convergence?
5. What tests are typically used to determine convergence of power series?

Tip:

Always consider both convergence and divergence points carefully to deduce the bounds for the radius of convergence!