Interval of Convergence with Center Calculator

What is Interval of Convergence?

For a power series centered at \( a \), the interval of convergence is the set of all \( x \) values for which the series converges. It can be one of three types:

• Finite interval: \( (a - R, a + R) \)
• Infinite interval: \( (-\infty, \infty) \)
• Semi-infinite interval: \( [a - R, a + R] \) or \( (a - R, a + R] \)

The radius of convergence \( R \) determines the width of the interval. The interval of convergence includes all \( x \) values within \( R \) units of the center \( a \).

How to Calculate Interval of Convergence

To find the interval of convergence for a power series:

1. Identify the center \( a \) of the power series
2. Apply the ratio test to find the radius of convergence \( R \)
3. Check the endpoints \( x = a + R \) and \( x = a - R \) for convergence
4. Combine the results to determine the interval of convergence

If the limit exists and is finite, \( R \) is the radius of convergence. If the limit is 0, the radius is infinite. If the limit does not exist, the radius is 0.

Example Calculation

Consider the power series \( \sum_{n=1}^{\infty} \frac{(x-3)^n}{n^2} \).

1. Identify the center: \( a = 3 \)
2. Apply the ratio test:
   \( \lim_{n \to \infty} \left| \frac{(x-3)^n / n^2}{(x-3)^{n+1} / (n+1)^2} \right| = \lim_{n \to \infty} \left| \frac{(x-3) (n+1)^2}{n^2} \right| \)
3. For convergence, the limit must be less than 1:
   \( \left| x - 3 \right| \lim_{n \to \infty} \left( \frac{n+1}{n} \right)^2 = \left| x - 3 \right| \leq 1 \)
4. Thus, the radius of convergence is \( R = 1 \)
5. Check endpoints:
   • At \( x = 4 \), the series converges
   • At \( x = 2 \), the series converges
6. Final interval of convergence: \( [2, 4] \)

Common Mistakes to Avoid

Always verify your calculations by testing specific values within the interval and checking the behavior at the endpoints.