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Interval of Convergence and Radius of Convergence Calculator

Reviewed by Calculator Editorial Team

This calculator helps you determine the interval and radius of convergence for power series. Understanding convergence is essential in calculus and analysis, as it tells us where a series converges to a finite value.

What is Convergence?

A power series is an infinite series of the form:

Σ (from n=0 to ∞) aₙ (x - c)ⁿ

The radius of convergence (R) is the distance from the center c where the series converges. The interval of convergence is the set of all x values where the series converges, typically expressed as [c - R, c + R].

There are three possible scenarios for convergence:

  1. The series converges only at x = c (R = 0).
  2. The series converges for all real numbers (R = ∞).
  3. The series converges for all x in the interval [c - R, c + R].

How to Find the Radius of Convergence

The most common method to find the radius of convergence is the Ratio Test. Here's how it works:

  1. Compute the limit: L = lim (n→∞) |aₙ₊₁ / aₙ|.
  2. If L < 1, the series converges absolutely.
  3. If L > 1, the series diverges.
  4. If L = 1, the test is inconclusive.

Once you find L, the radius of convergence is R = 1/L.

Note: The Ratio Test only gives the radius of convergence. You must perform additional tests to determine the interval of convergence.

Example Calculation

Consider the series Σ (from n=0 to ∞) (x - 2)ⁿ / n!. Let's find its radius of convergence.

  1. Identify aₙ = 1/n!.
  2. Compute the limit: L = lim (n→∞) |aₙ₊₁ / aₙ| = lim (n→∞) |(1/(n+1)!) / (1/n!)| = lim (n→∞) 1/(n+1) = 0.
  3. Since L = 0 < 1, the series converges absolutely for all x.
  4. Therefore, the radius of convergence is R = ∞.

The interval of convergence is all real numbers, (-∞, ∞).

Common Mistakes

When calculating convergence, avoid these common errors:

  • Assuming the Ratio Test gives the interval of convergence - it only gives the radius.
  • Forgetting to check the endpoints of the interval after finding the radius.
  • Incorrectly applying the Ratio Test to series that don't fit its conditions.

FAQ

What if the Ratio Test gives L = 1?
When L = 1, the Ratio Test is inconclusive. You must use another test, such as the Root Test or direct comparison, to determine convergence.
Can a power series have a finite radius of convergence?
Yes, many power series have finite radii of convergence. For example, the series Σ (xⁿ / n) has R = 1.
How do I know if a series converges at the endpoints?
After finding the radius, you must test the endpoints x = c - R and x = c + R separately using other convergence tests.