Interval of Convergence Power Series Calculator with Steps

What is Interval of Convergence?

The interval of convergence for a power series is the range of x-values for which the series converges. A power series is an infinite sum of terms of the form aₙ(x - c)ⁿ, where aₙ are coefficients and c is the center of the series.

There are three possible scenarios for the interval of convergence:

1. Converges only at x = c (radius of convergence = 0)
2. Converges for all real numbers (radius of convergence = ∞)
3. Converges on a finite interval (radius of convergence = R > 0)

How to Calculate Interval of Convergence

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

1. Identify the general form of the power series: Σ aₙ(x - c)ⁿ
2. Apply the Ratio Test to find the radius of convergence R
3. Check the endpoints of the interval (-R + c, R + c) to determine if the series converges at those points

Once you have the radius R, the potential interval of convergence is (c - R, c + R). You must then test the endpoints separately to see if they are included in the interval.

Example Calculation

Let's find the interval of convergence for the series Σ (n²xⁿ)/n³.

1. Rewrite the series in standard form: Σ (xⁿ)/n
2. Apply the Ratio Test:

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

3. The series converges when |x| < 1, so R = 1
4. Test the endpoints:
   • At x = 1: The series becomes Σ 1/n, which diverges (harmonic series)
   • At x = -1: The series becomes Σ (-1)ⁿ/n, which converges conditionally

The interval of convergence is [-1, 1).

Common Pitfalls

When calculating the interval of convergence, be aware of these common mistakes:

• Forgetting to test the endpoints of the interval
• Incorrectly applying the Ratio Test to the series
• Assuming the series converges at the endpoints when it actually diverges
• Miscounting the center of the series (c)