Series Calculator Interval of Convergence

What is Interval of Convergence?

The interval of convergence is the set of all real numbers for which a power series converges. A power series is an infinite sum of terms of the form \( a_n(x - c)^n \), where \( a_n \) are coefficients and \( c \) is the center of the series.

For a power series \( \sum_{n=0}^{\infty} a_n(x - c)^n \), the interval of convergence is typically expressed in one of three forms:

• An open interval \( (R - r, R + r) \)
• A single point \( \{R\} \)
• The entire real line \( (-\infty, \infty) \)

The radius of convergence \( r \) is the distance from the center \( c \) to the endpoints of the interval. The interval of convergence is centered at \( c \) and extends \( r \) units in both directions.

How to Calculate Interval of Convergence

To find the interval of convergence for a power series, follow these steps:

1. Identify the general form of the series: \( \sum_{n=0}^{\infty} a_n(x - c)^n \)
2. Apply the ratio test to find the radius of convergence \( r \):
   r = lim(n→∞) |aₙ / aₙ₊₁|
3. Check the endpoints \( x = c + r \) and \( x = c - r \) to determine if they are included in the interval.
4. Combine the results to form the interval of convergence.

Example Calculation

Consider the series \( \sum_{n=0}^{\infty} \frac{(x - 2)^n}{n!} \). Let's find its interval of convergence.

1. Identify the general form: \( a_n = \frac{1}{n!} \), \( c = 2 \)
2. Apply the ratio test:
   r = lim(n→∞) |(1/n!) / (1/(n+1)!)| = lim(n→∞) (n+1) = ∞
3. The series converges for all real numbers \( x \).
4. Therefore, the interval of convergence is \( (-\infty, \infty) \).

Common Pitfalls

When calculating the interval of convergence, be aware of these common mistakes:

• Assuming the series converges only at the center: The interval of convergence may include the center but extend beyond it.
• Forgetting to check the endpoints: The series may converge at one or both endpoints even if the radius of convergence is finite.
• Applying the wrong convergence test: The ratio test works for many power series, but other tests may be needed for more complex series.