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Interval of Convergence Calculator Reddit

Reviewed by Calculator Editorial Team

The interval of convergence is the set of all real numbers for which a power series converges. This calculator helps you determine the interval of convergence for any given power series, with visual representation and detailed explanation.

What is Interval of Convergence?

A power series is an infinite sum of terms that can be written in the form:

Σ (from n=0 to ∞) aₙ(x - c)ⁿ

The interval of convergence is the set of all x-values for which this series converges. It's typically expressed in one of three forms:

  1. An open interval (-R, R)
  2. A closed interval [-R, R]
  3. A half-open interval (-R, R] or [-R, R)

The radius of convergence (R) determines how far the series extends from the center point c. If R is infinite, the series converges for all real numbers.

How to Calculate Interval of Convergence

The standard method for finding the interval of convergence involves three steps:

  1. Find the radius of convergence using the ratio test
  2. Check for convergence at the endpoints
  3. Combine the results to determine the interval

The ratio test involves taking the limit of the absolute value of the ratio of consecutive terms as n approaches infinity. If this limit is less than 1, the series converges.

Formula

The radius of convergence R is found using the ratio test:

R = lim (n→∞) |aₙ / aₙ₊₁|

Once R is determined, the interval of convergence is:

(c - R, c + R)

You must then check the endpoints separately to determine if they should be included in the interval.

Worked Example

Consider the power series Σ (from n=0 to ∞) (x² - 4)ⁿ / (n² + 1).

Using the ratio test, we find:

R = lim (n→∞) |(x² - 4)² / (n² + 1) / (x² - 4)² / ((n+1)² + 1)| = lim (n→∞) (n² + 1) / ((n+1)² + 1) = 1

So the radius of convergence is 1. The potential interval is (-1, 1). Checking the endpoints:

  • At x = 1: The series becomes Σ (1 - 4)ⁿ / (n² + 1) = Σ (-3)ⁿ / (n² + 1), which diverges.
  • At x = -1: The series becomes Σ (1 - 4)ⁿ / (n² + 1) = Σ (-3)ⁿ / (n² + 1), which also diverges.

Therefore, the interval of convergence is (-1, 1).

Reddit Discussion Insights

Reddit users often discuss common pitfalls when calculating intervals of convergence:

  • Forgetting to check the endpoints separately
  • Miscounting the radius of convergence
  • Assuming the interval is always open when it might be closed
  • Misapplying the ratio test to series that don't satisfy its conditions

Many users find visual representations helpful, which is why our calculator includes a graph of the series behavior.

FAQ

What if the ratio test gives an indeterminate form?

If the ratio test results in an indeterminate form like 1/1, you may need to use another convergence test such as the root test or direct comparison.

Can a power series have an infinite radius of convergence?

Yes, if the limit in the ratio test is 0, the series converges for all real numbers, and the radius of convergence is infinite.

What does it mean if the interval of convergence is just a single point?

This means the power series only converges at its center point c, and nowhere else. The radius of convergence is 0 in this case.