Interval and Radius of Convergence for Power Series Calculator

A power series is an infinite sum of terms that are powers of a variable. The convergence of a power series determines where the series can be used to represent a function. This calculator helps you find both the radius and interval of convergence for any given power series.

What is Convergence in Power Series?

A power series converges if the sum of its terms approaches a finite limit as the number of terms increases. The convergence of a power series is determined by its radius of convergence, which is the distance from the center of the series where the series converges.

The interval of convergence is the set of all values of the variable for which the power series converges. It is typically expressed as an interval centered around the center of the series, with the radius of convergence as the distance from the center to the endpoints of the interval.

How to Find the Radius of Convergence

The radius of convergence (R) of a power series 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 as the number of terms approaches infinity.

Ratio Test Formula:

R = lim(n→∞) |aₙ / aₙ₊₁|

If the limit exists and is finite, then the radius of convergence is the reciprocal of the limit. If the limit is zero, the radius of convergence is infinite. If the limit is infinite, the radius of convergence is zero.

How to Find the Interval of Convergence

Once the radius of convergence is found, the interval of convergence can be determined by checking the endpoints of the interval centered around the center of the series. The endpoints are given by the center plus or minus the radius of convergence.

To find the interval of convergence, you need to check the convergence at the endpoints of the interval. This can be done using substitution or other convergence tests, such as the nth term test or the integral test.

Note: The interval of convergence may include one or both endpoints, depending on the behavior of the series at those points.

Worked Examples

Example 1: Finding the Radius of Convergence

Consider the power series: Σ (n=0 to ∞) (xⁿ)/n!

Using the ratio test:

lim(n→∞) |(xⁿ⁺¹)/(n+1)! / (xⁿ)/n!| = lim(n→∞) |x/(n+1)| = 0

Since the limit is zero, the radius of convergence is infinite. Therefore, the interval of convergence is (-∞, ∞).

Example 2: Finding the Interval of Convergence

Consider the power series: Σ (n=1 to ∞) (-1)ⁿ⁺¹ xⁿ / n³

Using the ratio test:

lim(n→∞) |[(-1)ⁿ⁺² xⁿ⁺¹ / (n+1)³] / [(-1)ⁿ⁺¹ xⁿ / n³]| = lim(n→∞) |-x n² / (n+1)³| = |x|

The radius of convergence is R = 1. To find the interval of convergence, we check the endpoints:

At x = 1: The series becomes Σ (-1)ⁿ⁺¹ / n³, which converges by the alternating series test.
At x = -1: The series becomes Σ (-1)ⁿ⁺² / n³, which also converges by the alternating series test.

Therefore, the interval of convergence is [-1, 1].

Frequently Asked Questions

What is the difference between radius and interval of convergence?: The radius of convergence is the distance from the center of the power series where the series converges. The interval of convergence is the set of all values of the variable for which the power series converges, which includes the radius of convergence and the endpoints of the interval.
How do I know if a power series converges?: A power series converges if the limit of the ratio of consecutive terms is less than 1. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the series may or may not converge, and additional tests are needed.
Can the interval of convergence be infinite?: Yes, if the radius of convergence is infinite, the interval of convergence is also infinite. This means the power series converges for all real numbers.
What happens if the radius of convergence is zero?: If the radius of convergence is zero, the power series only converges at the center of the series. This means the series does not converge for any other value of the variable.
How do I find the interval of convergence if the radius is finite?: To find the interval of convergence, you need to check the endpoints of the interval centered around the center of the series. The endpoints are given by the center plus or minus the radius of convergence. You can use substitution or other convergence tests to determine if the series converges at the endpoints.