The Radius of Convergence Calculator finds the distance from the center of a power series where the series converges. It uses the standard ratio/root tests to compute the radius and then states what that means for the interval of convergence.
Use it to quickly check whether your series converges near the point you expanded around, and to identify the endpoints you must test separately.
What “radius of convergence” means
A power series has the form:
\(\sum_{n=0}^{\infty} a_n (x-c)^n\), where \(c\) is the center.
The radius of convergence, written \(R\), is the distance from \(c\) to the nearest point where the series stops converging.
- If |x − c| < R, the series converges.
- If |x − c| > R, the series diverges.
- If |x − c| = R, the series may converge or diverge—test endpoints separately.
How the calculator computes \(R\)
The radius is determined by analyzing the coefficients \(a_n\). The two most common methods are the ratio test and the root test. The calculator supports the standard workflow: you provide a symbolic expression for the coefficient ratio (or the terms needed for the limit), and it converts the result into a numeric radius.
Method 1: Ratio test (typical form)
For many power series, you can compute:
\(L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|\).
Then:
\(R = \frac{1}{L}\), when \(L\neq 0\) and finite.
If \(L=0\), then \(R=\infty\). If \(L=\infty\), then \(R=0\).
Method 2: Root test (alternative)
You can also compute:
\(\rho = \limsup_{n\to\infty} \sqrt[n]{|a_n|}\).
Then:
\(R = \frac{1}{\rho}\), when \(\rho\neq 0\) and finite.
This is useful when \(a_n\) is easier to take roots of than to form a ratio.
Variables used in the calculator
| Symbol | Meaning | How it’s used |
|---|---|---|
| \(c\) | Center of the power series | Used to compute the interval endpoints \(c \pm R\). |
| Ratio limit (\(L\)) | Limit from the ratio test | Calculator uses \(R = 1/L\). |
| Radius (\(R\)) | Distance from center | Determines where the series converges. |
How to interpret the result
After computing \(R\), the convergence interval is:
- Open interval: (c − R, c + R)
- Endpoints: check separately by plugging into the series and using appropriate tests
If the calculator returns \(R=\infty\), the series converges for all real x. If it returns \(R=0\), it only converges at x=c (if it converges there).
Practical examples
Example 1: Quick radius from a ratio limit
Suppose your power series is centered at \(c=0\) and the coefficients satisfy:
\(\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = 2\).
Then the radius is:
\(R = \frac{1}{2} = 0.5\).
So the series converges for |x| < 0.5. You still test x=\pm 0.5 separately.
Example 2: Nonzero center changes only the endpoints
Now take the same ratio limit L=2, but expand about \(c=3\). The radius is still:
\(R=0.5\).
The convergence interval becomes (2.5, 3.5), and the endpoints are x=2.5 and x=3.5. The test at these endpoints depends on the specific series.
Common mistakes to avoid
- Forgetting endpoint tests: the ratio/root test does not guarantee convergence at |x − c| = R.
- Mixing up \(c\): radius is a distance; endpoints use c ± R.
- Using the wrong limit: the ratio limit should be computed for \(a_n\), not for the full term unless the form matches the ratio-test setup.
- Assuming units: mathematically, R is in the same units as x. The calculator treats it as a numeric distance; if your variables have units, interpret accordingly.
Frequently Asked Questions
What is the Radius of Convergence Calculator used for?
The Radius of Convergence Calculator computes the radius \(R\) for a power series \(\sum a_n(x-c)^n\). It uses the ratio or root-test limit to find how far from the center the series converges. It then reports the endpoint locations \(c \pm R\) and reminds you to test them.
Does the radius of convergence tell me whether the series converges at the endpoints?
No. The radius of convergence only guarantees convergence for points strictly inside \(|x-c|<R\) and divergence for \(|x-c|>R\). At \(|x-c|=R\), the test is inconclusive. You must plug in the endpoints and apply a series test.
What does it mean if the radius of convergence is infinity?
If \(R=\infty\), the series converges for every real value of \(x\). This happens when the coefficient growth is slow enough that the ratio limit is zero. Even then, you should still confirm convergence at any special points if your series is defined piecewise.
How do I find the interval of convergence from \(R\) and \(c\)?
Once you know \(R\) and the center \(c\), the interval is typically \((c-R,\,c+R)\). Those points satisfy \(|x-c|<R\). The endpoints \(x=c-R\) and \(x=c+R\) require separate endpoint tests using the original series.
Why do some problems require both ratio and root tests?
Some series make one limit easier than the other. If \(\left|\frac{a_{n+1}}{a_n}\right|\) is messy, using \(\sqrt[n]{|a_n|}\) may simplify. Both tests lead to the same radius when applied correctly. Use the one with the cleanest limit.
Next steps
Use the calculator to get \(R\) quickly, then verify the endpoints by substituting the boundary values into the series. If you share your series form (the \(a_n\) term and center \(c\)), you can also compute the ratio limit step-by-step and confirm the result.
