Interval of Convergence Calculator
Choose a power series pattern to find its radius and interval of convergence using the ratio test.
Choose a Series Pattern
Results
Radius of Convergence
1
Interval of Convergence
(-1, 1)
Number Line
Diverges at x = a − 1 because the terms do not approach 0.
Diverges at x = a + 1 because the terms do not approach 0.
This tool covers the most common series patterns used in Calculus II / AP Calculus. A custom series with non-polynomial coefficients needs a manual ratio-test check.
How the Interval of Convergence Calculator Works
Choose the power series pattern that matches your problem (these cover the most common forms seen in Calculus II and AP Calculus), enter the center a and any scale constant k, and the calculator applies the ratio test to find the radius of convergence, then checks both endpoints to determine whether they are included in the final interval.
Formula
The ratio test finds R from the limit of |term(n+1)/term(n)| as n approaches infinity, requiring that limit to be less than 1 for convergence. Each endpoint x = a - R and x = a + R is then tested separately, since the ratio test is inconclusive exactly at the boundary.
Example
For the series sum of (x-a)^n / n starting at n=1, the ratio test gives radius of convergence R = 1. At x = a + 1 the series becomes the harmonic series, which diverges. At x = a - 1 it becomes the alternating harmonic series, which converges. The interval of convergence is [a - 1, a + 1).
Limitations
This tool covers the classic power series forms used in most calculus courses. Series that do not match one of the listed patterns, such as those with non-polynomial coefficients or mixed terms, require applying the ratio test and endpoint checks by hand or with a computer algebra system.
How to use the result responsibly
- Check that every input uses the unit and time period shown beside the field.
- Compare the result with the worked example and formula above.
- For financial, tax, health, or legal decisions, confirm current rules with a qualified professional or the relevant authority.
Frequently Asked Questions
What happens at the endpoints of the interval of convergence?
The ratio test cannot determine convergence exactly at x = a - R or x = a + R, so each endpoint must be substituted into the original series and tested separately, often with the alternating series test or a p-series comparison.
What does a radius of convergence of 0 mean?
It means the series only converges at its center point x = a and diverges everywhere else.
What does a radius of convergence of infinity mean?
It means the series converges for every real number x, which is common for series like the Taylor series of e^x.
Editorially reviewed September 22, 2026. Read our calculation methodology and editorial policy.