Interval of Power Series Calculator

What is the Interval of a Power Series?

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 a constant. The interval of convergence is the set of all x-values for which the series converges to a finite value.

There are three possible outcomes for the interval of convergence:

1. The series converges only at \( x = c \) (radius of convergence is 0).
2. The series converges for all real numbers (radius of convergence is infinite).
3. The series converges for all \( x \) in an interval \( (c - R, c + R) \), where \( R \) is the radius of convergence.

The interval of convergence is often expressed as \( (c - R, c + R) \), but it may be a subset of this interval if the series converges only at one or both endpoints.

How to Calculate the Interval of Convergence

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

1. Identify the general form of the power series: \( \sum_{n=0}^{\infty} a_n(x - c)^n \).
2. Apply the Ratio Test to find the radius of convergence \( R \).
3. Check the endpoints \( x = c - R \) and \( x = c + R \) to determine if the series converges at these points.
4. Combine the results to form the interval of convergence.

The Ratio Test is the most common method for finding the radius of convergence. It involves taking the limit of the absolute value of the ratio of consecutive terms as \( n \) approaches infinity.

The Formula

The radius of convergence \( R \) is given by:

Where \( a_n \) are the coefficients of the power series. The interval of convergence is then \( (c - R, c + R) \), possibly including one or both endpoints.

After finding \( R \), you must check the endpoints \( x = c - R \) and \( x = c + R \) separately to determine if they are included in the interval of convergence.

Worked Example

Consider the power series \( \sum_{n=0}^{\infty} \frac{(x - 2)^n}{n!} \).

1. Identify the general form: \( a_n = \frac{1}{n!} \), \( c = 2 \).
2. Apply the Ratio Test:
   R = lim (n→∞) |(n!)/( (n+1)! )| = lim (n→∞) 1/(n+1) = 0
3. Since \( R = 0 \), the series converges only at \( x = 2 \).

The interval of convergence for this series is \( \{2\} \).

Interpreting the Results

The interval of convergence tells you where the power series converges to a finite value. If the radius of convergence is finite, the series converges for all \( x \) within \( R \) units of \( c \).

If the series converges at one or both endpoints, those points are included in the interval. If it diverges at both endpoints, the interval is open.

For practical purposes, you can use the interval of convergence to determine where a power series provides a good approximation to the function it represents.