To analyze the convergence of the power series $\sum a_n (x - 3)^n$, we’ll use the information provided: it converges at $x = -1$ and diverges at $x = 10$. This helps establish the interval of convergence of the series around the center $x = 3$.

The convergence of a power series depends on its radius of convergence $R$. The radius $R$ is determined by the distance between $x = 3$ (the center) and the points where convergence and divergence behaviors are known:

• Convergence at $x = -1$ suggests that $|x - 3| \leq R$, so $R \geq |(-1) - 3| = 4$.
• Divergence at $x = 10$ implies that $|x - 3| > R$, so $R < |10 - 3| = 7$.

Thus, we conclude that the radius of convergence $R$ lies between 4 and 7: $4 \leq R < 7$

Now, let’s examine the specific points given:

(a) $x = 3$

For $x = 3$:

• $|x - 3| = 0$, which is within any radius $R > 0$.
• Therefore, the series converges at $x = 3$.

(b) $x = 6$

For $x = 6$:

• $|x - 3| = 3$, which is less than both 4 and 7.
• This distance is within the radius of convergence, so the series converges at $x = 6$.

(c) $x = -5$

For $x = -5$:

• $|x - 3| = 8$, which is greater than both 4 and 7.
• Since this distance exceeds the radius of convergence, the series diverges at $x = -5$.

Summary of Results:

1. (a) $x = 3$: Converges.
2. (b) $x = 6$: Converges.
3. (c) $x = -5$: Diverges.

Would you like further details on power series convergence, or do you have any questions?

Related Questions

1. How is the radius of convergence determined for a general power series?
2. What does it mean for a series to converge absolutely versus conditionally?
3. How can the Ratio Test be used to find the radius of convergence?
4. What is the interval of convergence for a series centered at a point other than zero?
5. How does the behavior at the endpoints of an interval of convergence vary?

Tip:

For power series, knowing one point of convergence and one point of divergence can significantly narrow down the possible radius of convergence.