Suppose that ∑∞*𝑛*=0*𝑎𝑛𝑥𝑛*∑n=0∞anxn converges for *𝑥*=2x=2 and diverges for *𝑥*=−4.x=−4. What can be said about the convergence or divergence of the following series?

∑∞*𝑛*=02*𝑛𝑎𝑛*and∑∞*𝑛*=0(−1)*𝑛*5*𝑛𝑎𝑛*


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.

Explore the convergence and divergence of power series at specific points. Understand how to determine the radius of convergence and apply it to series evaluation. Detailed analysis of series behavior at x = 2 and x = -4 provided.

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