Interval of Convergence Calculator
An interval of convergence calculator is a calculus tool that determines the full set of real numbers for which a power series converges. It first computes the radius of convergence using the Ratio Test and then evaluates the behavior of the series at the boundary endpoints by substituting their values into the general term. By evaluating the stabilization of numerical partial sums up to one thousand terms, the tool decides whether each endpoint converges or diverges. Students use this calculator to identify if intervals are open, closed, or half-open.
Determine the interval of convergence for a power series. Specify the general term and center to find the radius and check the endpoints.
Quick Answer
Find the complete interval of convergence for a power series, including boundary endpoints. Enter the general term and center point to compute the bracketed interval.
Must include variables x and n.
Try Examples
Endpoint Convergence Testing (Heuristic)
Left Endpoint (c - R): x = -1.007 (diverges)
Term a_1000 = 1.070214E0
Right Endpoint (c + R): x = 1.007 (diverges)
Term a_1000 = 1.070214E0
Interval =
(-1.007, 1.007)
Radius of Convergence R = 1.007
Notice: Endpoint testing is performed using a numerical partial sum heuristic. For rigorous mathematical proof, manual verification is highly recommended.
Example: For a_n = x^n/n centered at 0, the interval of convergence is [-1, 1).
Examples
x^n / n
Interval = [-1, 1)
x^n / n^2
Interval = [-1, 1]
x^n
Interval = (-1, 1)
(-1)^n * (x-3)^n / n
Interval = (2, 4]
How it works
The calculator uses the Ratio Test to find the radius of convergence R, then substitutes the endpoint values x = c − R and x = c + R back into the series to test for convergence:
Possible Interval Formats
- Open: (c - R, c + R)
- Closed: [c - R, c + R]
- Half-open: [c - R, c + R) or (c - R, c + R]
Heuristic Nature of Numerical Convergence
Testing convergence numerically by summing the first 1000 terms is a heuristic approach. For extremely slowly diverging series (like the sub-harmonic series Σ 1/(n ln(n))) or highly oscillatory series, 1000 terms might not be sufficient to show divergence. Always supplement this calculator with analytic checks for complete mathematical certainty.
Related Series Tools
For general series convergence testing (not involving variables x), visit our Convergence Calculator. To compute the radius of convergence alone, use the Radius of Convergence Calculator.
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