Radius of Convergence Calculator

Introduction: What the radius of convergence means for a power series

A power series can look harmless term by term, but its convergence is governed by how its coefficients grow as n increases.

f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n.

The radius of convergence, usually written as R, is the distance from the center of the series to the edge of the largest open disc on which the sum converges absolutely. In practice, that means the series converges for all complex numbers z with |z| < R and diverges for all |z| > R. On the boundary |z| = R, some points may converge while others do not, so the edge always needs separate checking.

Knowing the radius of convergence helps when you are working with Taylor or Maclaurin series, checking analytic continuation, or using a power series as a numerical approximation.

Root test and ratio test formulas for radius of convergence

Two standard tests appear again and again when you want the radius of convergence of a power series.

Root test formula

For a series

\sum_{n=0}^{\infty} a_n z^n,

the root test gives the radius of convergence as

In plain words: take the limit superior of the sequence |a_n|^{1/n} as n goes to infinity, and then take the reciprocal. That reciprocal is the radius of convergence. If the limit superior is zero, the radius is infinite; if the limit superior is infinite, the radius is zero.

Ratio test formula

When the ratios of consecutive coefficients behave nicely, the ratio test is often easier to apply. If the limit

L = lim_{n \to \infty} |a_{n+1} / a_n|

exists, then the radius of convergence is

R = 1 / L (with the usual conventions: if L = 0, then R = \infty; if L = \infty, then R = 0).

That is why textbook exercises on power series usually point you to either the root test or the ratio test, depending on whether the coefficients are easier to read through their overall growth or through their term-to-term ratios.

How this radius of convergence calculator estimates R

Because real-world coefficient lists are finite, this calculator treats the values you enter as a numerical snapshot of the infinite power series.

Suppose you enter coefficients up to index N, so you have

a_0, a_1, a_2, \dots, a_N.

The calculator performs the following steps:

1. For each index k with a nonzero coefficient, it computes the absolute value |a_k|.
2. It then computes the k-th root |a_k|^{1/k} for k \ge 1.
3. It looks at the largest of these roots over the available indices, which acts as a finite-sample approximation to the limit superior.
4. Finally, it takes the reciprocal of this maximum root to produce an estimate for the radius of convergence R.

As you supply more coefficients, this finite approximation usually moves closer to the true radius, especially when the coefficients follow a regular pattern such as geometric growth, factorial growth, or another recurrence that is easy to sample.

How to use this radius of convergence calculator

Use this calculator when you already have coefficients for a power series centered at 0 and want a quick estimate of the convergence radius.

The input field accepts a comma-separated list of coefficients for a power series centered at 0:

\sum_{n=0}^{\infty} a_n z^n.

1. Enter coefficients in order, starting from a_0. Use commas to separate them, for example:
   1, -1/2, 1/3, -1/4.
2. Use fractions or decimals. The parser accepts simple fractions like 1/3, integers, and decimal numbers like 0.25.
3. Run the calculation with the provided button. The tool interprets the list as a_0, a_1, a_2, \dots, applies the numerical root-test approach, and outputs an estimated radius of convergence R.
4. Interpret the result as the approximate distance from 0 to the edge of the disc where the series converges absolutely. If the estimated R is large, the disc of convergence is wide; if it is small, the disc is narrow.

Interpreting a radius of convergence estimate

Once the calculator returns R, it helps to translate that number back into the geometry of the power series.

• Disc of convergence: The series converges absolutely for all complex numbers with |z| < R.
• Interval of convergence (real variable): If you restrict the same power series to real x, it converges for |x| < R. The endpoints x = \pm R must be checked separately, because the boundary may behave differently from the interior. The set of real x values that converge is the interval of convergence, while R is the radius measured from the center.
• Singularities: For analytic functions, the radius of convergence matches the distance from the expansion center to the nearest singularity in the complex plane. That gives you a quick hint about where the function can be continued analytically.

Worked radius of convergence examples

The examples below show how a radius of convergence estimate behaves for a series with constant coefficients and for one whose coefficients explode factorially.

Example 1: Geometric series

Consider the geometric series

\sum_{n=0}^{\infty} z^n = 1 + z + z^2 + z^3 + \cdots.

Here, all coefficients are a_n = 1. The exact theory says this series converges if and only if |z| < 1, so the true radius of convergence is R = 1.

To approximate this using the calculator, you could enter a finite list like

1, 1, 1, 1, 1, 1, 1, 1.

For each nonzero coefficient, we have |a_k|^{1/k} = 1^{1/k} = 1, so the maximum root is 1, and the estimated radius is its reciprocal, which is also 1. In this case, the numerical method recovers the exact value even from a short coefficient list.

Example 2: Factorial growth

Now look at the series

\sum_{n=0}^{\infty} n! z^n.

The coefficients grow very quickly: a_n = n!. Using the ratio test, we have

|a_{n+1} / a_n| = (n+1)! / n! = n+1, which tends to infinity as n \to \infty. Therefore, L = \infty and the exact radius of convergence is R = 0. The series diverges for every nonzero z.

If you enter a few terms, for example

1, 1, 2, 6, 24, 120

corresponding to 0!, 1!, 2!, 3!, 4!, 5!, the calculator computes |a_k|^{1/k} for k \ge 1. These roots get larger as k increases, and the maximum root becomes quite big even for a short list. The reciprocal of that maximum is then very small, providing a numerical estimate that the radius is near zero, in agreement with the exact theory.

Comparison: root test vs ratio test for power series

When you are estimating the radius of convergence of a power series, the root test and ratio test often point to the same answer from different angles.

Radius of convergence limitations and assumptions

This calculator is best treated as a numerical guide for a power series, not as a substitute for a proof.

• Finite data: The estimate uses only the coefficients you enter. With a short list, the approximation to the limit superior may be crude. Adding more terms generally improves the estimate.
• Irregular coefficients: If the coefficients oscillate wildly or have no stable growth pattern, the finite maximum of |a_k|^{1/k} may not reflect the true long-term behavior. The estimate can then be misleading.
• Zero coefficients: When early coefficients are zero, the root-test part still works on later terms, but the ratio-test line may skip gaps because a zero term makes a consecutive ratio meaningless. This is usually harmless, but it can leave the ratio estimate unavailable.
• Approximate lim sup: Mathematically, the radius depends on the limit superior as n tends to infinity. The calculator replaces this with a maximum over finitely many terms, so it always produces an approximation, not a guarantee.
• Center at zero: The input is interpreted as coefficients of a series centered at 0. If your series is centered at another point z_0, you can still use the tool by shifting variables or by remembering that the disc of convergence is centered at z_0 with radius R.
• Absolute convergence only: The radius describes where the series converges absolutely. It does not, by itself, tell you about conditional convergence at the boundary, which must be analyzed separately.

If the numerical result conflicts with what you expect from theory, double-check the coefficients, add more terms if possible, and compare the output against the exact root or ratio test by hand.

Who benefits from a radius of convergence calculator

This radius of convergence calculator is useful whenever you want a fast check on a series written by hand, generated by a recurrence, or sampled from another computation.

• Experiment with different coefficient patterns and see how they affect the radius of convergence.
• Build intuition about the difference between radius and interval of convergence.
• Check numerical series coming from data or algorithms when a closed-form solution is unavailable.

Used alongside formal methods such as the root test, the ratio test, or knowledge of singularities, it provides quick feedback on how far a power series can be trusted before it stops converging.