Radius and Interval Convergence Calculator

This calculator helps determine the radius and interval of convergence for power series. Understanding convergence is essential in calculus and analysis, as it defines the range of values for which an infinite series converges to a finite limit.

What is Convergence?

Convergence refers to the behavior of an infinite series as the number of terms approaches infinity. A series converges if the sequence of its partial sums approaches a finite limit. The radius of convergence is the distance from the center of the power series within which the series converges.

The interval of convergence is the range of x-values for which the series converges. It can be expressed as (a - R, a + R), where a is the center of the series and R is the radius of convergence.

How to Calculate Radius and Interval of Convergence

The radius of convergence can be found using the ratio test or the root test. The most common method is the ratio test, which involves taking the limit of the absolute value of the ratio of consecutive terms.

Ratio Test Formula

For a power series Σ aₙ(x - a)ⁿ, the radius of convergence R is given by:

R = lim (n→∞) |aₙ / aₙ₊₁|

If the limit is L, then R = 1/L if L ≠ 0, and R = ∞ if L = 0.

Once the radius is determined, the interval of convergence must be checked at the endpoints a - R and a + R to determine if the series converges at these points.

Example Calculation

Consider the power series Σ (x - 2)ⁿ / n!.

Using the ratio test:

lim (n→∞) |(x - 2)ⁿ / n!| / |(x - 2)ⁿ⁺¹ / (n + 1)!| = lim (n→∞) |(x - 2)| / (n + 1) = 0 for all x.

This means the series converges for all x, so the radius of convergence is ∞ and the interval of convergence is (-∞, ∞).

FAQ

What is the difference between radius and interval of convergence?: The radius of convergence is the distance from the center of the power series within which the series converges. The interval of convergence is the range of x-values for which the series converges, which may include or exclude the endpoints.
How do I know if a series converges at the endpoints?: After finding the radius of convergence, you must test the endpoints separately using other convergence tests such as the nth term test or direct comparison.
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 has a radius of convergence of 1.
What if the ratio test gives an indeterminate form?: If the ratio test results in an indeterminate form, you may need to use the root test or other convergence tests to determine the radius of convergence.