Given the information that the power series $\sum_{n=0}^{\infty} a_n x^n$ converges at $x = 2$ and diverges at $x = -4$, we can analyze the convergence of the following series:

1. Analyzing the Radius of Convergence

As before, the series $\sum_{n=0}^{\infty} a_n x^n$ has a radius of convergence $R$. Since the series converges at $x = 2$ and diverges at $x = -4$, the radius of convergence $R$ satisfies: $2 \leq R < 4$ This means that the series converges for $|x| < R$ and diverges for $|x| > R$.

2. Convergence of $\sum_{n=0}^{\infty} 2^n a_n$

To determine the convergence of the series $\sum_{n=0}^{\infty} 2^n a_n$, we substitute $x = 2$ into the original series $\sum_{n=0}^{\infty} a_n x^n$. This transforms the series into: $\sum_{n=0}^{\infty} a_n 2^n$ This is equivalent to evaluating the original series at $x = 2$. Since the series converges at $x = 2$, the series $\sum_{n=0}^{\infty} 2^n a_n$ converges.

3. Convergence of $\sum_{n=0}^{\infty} (-1)^n 5^n a_n$

To determine the convergence of the series $\sum_{n=0}^{\infty} (-1)^n 5^n a_n$, we substitute $x = -5$ into the original series $\sum_{n=0}^{\infty} a_n x^n$. This transforms the series into: $\sum_{n=0}^{\infty} a_n (-5)^n$ This is equivalent to evaluating the original series at $x = -5$. Since $|-5| = 5$ is greater than the radius of convergence $R$, the series $\sum_{n=0}^{\infty} (-1)^n 5^n a_n$ diverges.

Summary

• $\sum_{n=0}^{\infty} 2^n a_n$ converges.
• $\sum_{n=0}^{\infty} (-1)^n 5^n a_n$ diverges.

Would you like more details or have any further questions?

Relative Questions:

1. What methods can be used to determine the radius of convergence for a given power series?
2. How does the ratio test apply to the convergence of power series?
3. What role does the endpoint play in determining the convergence of a power series?
4. Can a power series converge conditionally at the boundary of its radius of convergence?
5. How does the behavior of a power series at the boundary affect its applications in real analysis?

Tip:

When analyzing series, especially with varying powers of $x$, consider substituting specific values into the series to check for convergence or divergence based on the radius of convergence.