What Is the Interval of Convergence?
A power series Σ c(n) (x − a)ⁿ is a polynomial of infinite degree, and like any infinite sum it converges for some inputs and diverges for others. The set of x values where it converges is always an interval centered at a, called the interval of convergence. Its half-width is the radius of convergence R, and the whole story reduces to two questions: how big is R, and what happens at the two endpoints x = a ± R.
The radius comes straight from the ratio test. The series converges when the ratio of successive terms stays below one in size, which gives
R = 1 / lim |c(n+1) / c(n)|
The endpoints are the delicate part, because the ratio test falls silent exactly at the boundary and each end must be checked on its own with a separate convergence test.
How to Use the Interval of Convergence Calculator
Enter the coefficient formula c(n) and the center a. The calculator estimates the limiting coefficient ratio for large n, converts it to the radius R, and then probes both endpoints numerically by examining whether the terms vanish and whether the partial sums settle. It reports the interval with the correct bracket at each end, a square bracket where the endpoint converges and a parenthesis where it diverges, and plots a partial sum so you can see the polynomial approximation behaving inside the interval and breaking down outside it.
Worked Example
The series Σ xⁿ/n has c(n) = 1/n. The coefficient ratio is n/(n+1), which tends to 1, so R = 1 and the interval is centered on 0 with endpoints at ±1. Now the endpoints, which is where this series earns its fame:
At x = 1 the series is Σ 1/n, the harmonic series, which diverges. At x = −1 it is Σ (−1)ⁿ/n, the alternating harmonic series, which converges.
So the interval of convergence is [−1, 1), closed on the left, open on the right. The same radius, two different verdicts at the two ends, decided by two different tests. That asymmetry is the whole lesson of endpoint analysis, and it is why the boundary can never be settled by the radius alone.
The Three Possible Shapes
Every power series lands in one of three cases. When the coefficient ratio tends to zero, R is infinite and the series converges for all real x, as with the exponential series Σ xⁿ/n! whose factorial denominators crush every ratio. When the ratio grows without bound, R is zero and the series converges only at its center, useless as a function but a genuine possibility. In the generic middle case R is a finite positive number and the interval spans from a − R to a + R, with each endpoint independently open or closed. Inside the interval the series converges absolutely and defines a smooth function you may differentiate and integrate term by term; outside it the terms fail to shrink and the sum is meaningless. The radius of convergence tool focuses on finding R, while this page completes the picture with the endpoints.
Common Mistakes to Avoid
- Stopping at the radius. R names an open interval, but the endpoints must be tested separately. A complete answer states the bracket at each end, and the two ends frequently disagree, as the harmonic example shows.
- Using the ratio test at the endpoints. At x = a ± R the ratio test gives exactly 1 and says nothing. Switch to the p-series, alternating series, or divergence test there instead.
- Forgetting to shift by the center. For a series in (x − a), the interval is centered at a, not at 0. The endpoints are a − R and a + R, so a series centered at 1 with radius 1/2 lives on (1/2, 3/2), a slip that costs many exam points.
- Trusting numerics at a borderline endpoint. Slowly diverging series like the harmonic one look almost convergent to a finite computation. Numeric endpoint checks are a strong hint, but a rigorous answer names the analytic test that settles the boundary.
Real-World Applications
Power series are how computers and calculators evaluate the functions you take for granted. A sine, a logarithm, or an exponential is computed from its series, and the interval of convergence tells the algorithm which inputs the series can handle directly and which must first be reduced into range by identities. Range reduction, the trick of using periodicity or scaling to pull an argument into the convergent zone, exists precisely because series do not converge everywhere.
In physics and engineering, series solutions to differential equations are only valid inside their radius of convergence, which often marks a genuine physical boundary such as a singularity in a field or a resonance in a circuit. Perturbation methods across quantum mechanics, fluid dynamics, and control theory expand quantities in a small parameter, and the radius of convergence sets how large that parameter may grow before the expansion collapses. Signal processing and complex analysis lean on the same idea through the closely related radius of a Taylor or Laurent series, where the distance to the nearest singularity in the complex plane turns out to equal the radius of convergence, a beautiful link this real-variable page quietly rests on.
Frequently Asked Questions
Why is the convergence set always an interval?
Because convergence depends only on the size |x − a|. If the series converges at some distance from the center, it converges at every smaller distance too, which forces the convergence set to be a symmetric interval around a. Only the endpoints, at the exact boundary distance, can behave irregularly.
Can the two endpoints really behave differently?
Yes, and often they do. The series Σ xⁿ/n converges at −1 but not at 1. One endpoint can converge while the other diverges, both can converge, or both can diverge, so each must be tested on its own with an appropriate series test.
What does a radius of zero mean?
The series converges only at its center a and nowhere else, so it fails to define a function on any interval. This happens when coefficients grow very fast, such as c(n) = n!, and it signals that the series is a formal object rather than a usable expansion.
How is this different from the radius of convergence tool?
The radius tool answers only how wide the interval is. This page adds the endpoint analysis that turns the open interval (a − R, a + R) into the exact set, deciding whether each boundary point belongs. The radius is half the answer; the endpoints complete it.
Can I use the root test instead of the ratio test?
Yes. The root test gives R = 1 / lim |c(n)|^(1/n) and can succeed on coefficients where the ratio test struggles, such as those defined piecewise or with alternating structure. The two tests always agree on the radius when both apply, and the endpoints still need separate handling either way.
Why does the graph diverge outside the interval?
Beyond the endpoints the terms c(n)(x − a)ⁿ stop shrinking and instead grow, so partial sums swing wildly rather than settling. The plot shows the partial sum tracking a smooth function inside the interval and then flying apart just past the boundary, the visual signature of divergence.