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Interval of Convergence for A Series Calculator

Reviewed by Calculator Editorial Team

The interval of convergence for a series is the set of all real numbers for which the series converges. This calculator helps determine the interval of convergence for power series, which are series of the form Σaₙxⁿ where n ranges from 0 to infinity.

What is Interval of Convergence?

The interval of convergence for a power series is the set of all x-values for which the series converges. It's typically expressed in the form (a, b), where a and b are real numbers, and the series converges for all x in this interval.

There are three possible scenarios for the interval of convergence:

  1. The series converges only at x = 0.
  2. The series converges for all real numbers (the interval is (-∞, ∞)).
  3. The series converges for a finite interval (a, b).

Note: The interval of convergence is always a finite interval or the entire real line. It cannot be an infinite interval in the sense of extending to infinity in one direction.

How to Calculate Interval of Convergence

To determine the interval of convergence for a power series Σaₙxⁿ, follow these steps:

  1. Find the radius of convergence (R) using the ratio test or the root test.
  2. Check for convergence at the endpoints x = R and x = -R.
  3. Combine the results to form the interval of convergence.

Using the Ratio Test:

lim (n→∞) |aₙ₊₁ / aₙ| = L

If L < 1, the series converges absolutely for |x| < R.

If L > 1, the series diverges for all x ≠ 0.

If L = 1, the test is inconclusive.

The radius of convergence R is given by 1/L when L is finite and positive.

Examples of Calculating Interval of Convergence

Example 1: Series Σ (n²xⁿ)/n!

Using the ratio test:

lim (n→∞) |(n+1)²xⁿ⁺¹ / (n+1)!| / |n²xⁿ / n!| = lim (n→∞) (n+1)²x / (n+1) = |x|

For convergence, |x| < 1, so R = 1.

Check endpoints:

  • At x = 1: The series becomes Σ (n²)/n! which converges.
  • At x = -1: The series becomes Σ (-1)ⁿ(n²)/n! which converges.

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

Example 2: Series Σ (xⁿ)/n³

Using the ratio test:

lim (n→∞) |xⁿ⁺¹ / (n+1)³| / |xⁿ / n³| = lim (n→∞) |x| / (1 + 1/n)³ = |x|

For convergence, |x| < 1, so R = 1.

Check endpoints:

  • At x = 1: The series becomes Σ 1/n³ which converges.
  • At x = -1: The series becomes Σ (-1)ⁿ/n³ which converges.

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

FAQ

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

The radius of convergence is the distance from the center of the power series (usually x=0) within which the series converges. The interval of convergence includes the radius of convergence and may also include the endpoints where the series converges.

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

You need to check the convergence at the endpoints separately using other tests like the nth-term test or comparison test, since the ratio and root tests are inconclusive at the endpoints.

Can the interval of convergence be infinite?

No, the interval of convergence is always finite or the entire real line. It cannot be infinite in one direction.