State The Interval and Radius of Convergence Calculator

What is Convergence?

The convergence of a power series is a fundamental concept in calculus that determines the range of values for which the series converges to a finite limit. For a power series centered at \( a \),

The radius of convergence \( R \) is the distance from the center \( a \) within which the series converges. The interval of convergence is the interval \( (a - R, a + R) \) where the series converges.

There are three possible scenarios for the interval of convergence:

1. The series converges only at \( x = a \) (radius of convergence \( R = 0 \)).
2. The series converges for all real numbers \( x \) (radius of convergence \( R = \infty \)).
3. The series converges for \( x \) in the open interval \( (a - R, a + R) \), and may or may not converge at the endpoints.

How to Calculate Convergence

To determine the radius and interval of convergence, follow these steps:

1. Identify the power series: Write the series in the standard form \( \sum_{n=0}^{\infty} c_n (x - a)^n \).
2. Apply the Ratio Test: Compute the limit \( L = \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right| \).
3. Determine the radius: The radius of convergence is \( R = \frac{1}{L} \) if \( L \neq 0 \). If \( L = 0 \), the radius is \( R = \infty \). If \( L = \infty \), the radius is \( R = 0 \).
4. State the interval: The interval of convergence is \( (a - R, a + R) \).
5. Check endpoints: Use additional tests (Ratio Test, Root Test, or direct substitution) to determine if the series converges at \( x = a + R \) and \( x = a - R \).

Example Calculation

Consider the power series \( \sum_{n=0}^{\infty} \frac{(x - 3)^n}{n!} \).

1. Identify the series: \( c_n = \frac{1}{n!} \), \( a = 3 \).
2. Apply the Ratio Test:
   \( L = \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right| = \lim_{n \to \infty} \left| \frac{1/(n+1)!}{1/n!} \right| = \lim_{n \to \infty} \frac{1}{n+1} = 0 \)
3. Determine the radius: Since \( L = 0 \), the radius of convergence is \( R = \infty \).
4. State the interval: The series converges for all real numbers \( x \).

The interval of convergence is \( (-\infty, \infty) \).