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

1. Analyzing the Radius of Convergence

The power series $\sum_{n=0}^{\infty} a_n x^n$ has a radius of convergence $R$. For the series to converge at $x = 2$ and diverge at $x = -4$, the radius of convergence $R$ must satisfy: $2 \leq R < 4$ This is because the series converges when $|x| < R$ and diverges when $|x| > R$. Since $|2| = 2$ is within the radius of convergence, the series converges at $x = 2$. Conversely, since $|-4| = 4$ is beyond the radius of convergence, the series diverges at $x = -4$.

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 can 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 the same as evaluating the original series at $x = -2$. Since $|-2| = 2$ is within the radius of convergence $R$, the series $\sum_{n=0}^{\infty} (-2)^n a_n$ converges.

3. Convergence of $\sum_{n=0}^{\infty} (-1)^n 7^n a_n$

To determine the convergence of the series $\sum_{n=0}^{\infty} (-1)^n 7^n a_n$, we substitute $x = -7$ into the original series $\sum_{n=0}^{\infty} a_n x^n$. This transforms the series into: $\sum_{n=0}^{\infty} a_n (-7)^n$ This is equivalent to evaluating the original series at $x = -7$. Since $|-7| = 7$ is greater than the radius of convergence $R$, the series $\sum_{n=0}^{\infty} (-1)^n 7^n a_n$ diverges.

Summary

• $\sum_{n=0}^{\infty} (-2)^n a_n$ converges.
• $\sum_{n=0}^{\infty} (-1)^n 7^n a_n$ diverges.

Would you like more details on this analysis or have any questions?

Relative Questions:

1. How do you determine the radius of convergence for a power series?
2. What is the difference between absolute and conditional convergence?
3. How does the root test relate to the radius of convergence?
4. What is the significance of the radius of convergence in practical applications?
5. Can a power series converge outside its radius of convergence under any circumstances?

Tip:

When working with power series, always pay attention to the radius of convergence, as it plays a crucial role in determining where the series converges and where it diverges.