Power Series Interval Calculator

A power series is an infinite sum of terms that are powers of a variable. The interval of convergence is the set of all values for which the series converges. This calculator helps determine the interval of convergence for a given power series.

What is a Power Series?

A power series is a mathematical series of the form:

f(x) = Σ (from n=0 to ∞) aₙ(x - c)ⁿ

Where:

aₙ are coefficients
c is the center of the series
x is the variable

Power series are fundamental in calculus and analysis, providing a way to represent functions as infinite sums of simpler terms.

Interval of Convergence

The interval of convergence is the set of all x-values for which the power series converges. It can be determined using the ratio test:

lim (n→∞) |aₙ₊₁ / aₙ| = L

If L < 1, the series converges absolutely.

If L > 1, the series diverges.

If L = 1, the test is inconclusive.

The radius of convergence (R) is given by:

R = 1 / L

The interval of convergence is then:

(c - R, c + R)

Endpoints may need to be tested separately using other convergence tests.

How to Use This Calculator

To use the power series interval calculator:

Enter the coefficients of your power series in the input fields
Specify the center of the series (c)
Click "Calculate" to determine the interval of convergence
Review the results and chart visualization

Note: This calculator assumes the series is centered at x = c. For series not centered at zero, adjust the coefficients accordingly.

Worked Examples

Example 1: Basic Power Series

Consider the series Σ (from n=0 to ∞) (x/2)ⁿ

Using the ratio test:

lim (n→∞) |(x/2)ⁿ⁺¹ / (x/2)ⁿ| = |x/2| = L

For convergence, L < 1 → |x| < 2 → -2 < x < 2

The interval of convergence is (-2, 2).

Example 2: Series with Different Center

Consider the series Σ (from n=0 to ∞) (x-3)ⁿ

Using the ratio test:

lim (n→∞) |(x-3)ⁿ⁺¹ / (x-3)ⁿ| = |x-3| = L

For convergence, L < 1 → |x-3| < 1 → 2 < x < 4

The interval of convergence is (2, 4).

Frequently Asked Questions

What is the difference between radius of convergence and interval of convergence?

The radius of convergence is the distance from the center of the series where the series converges. The interval of convergence includes the center and extends this distance in both directions, plus any endpoints where the series may converge.

How do I know if the series converges at the endpoints?

You must test the endpoints separately using other convergence tests like the nth term test or integral test, as the ratio test is inconclusive at the endpoints.

What if the ratio test gives L = 1?

When L = 1, the ratio test is inconclusive, and you must use another test to determine convergence at that point.