Radius of Convergence Calculator

What is the radius of convergence?

Many functions in calculus and complex analysis can be written as power series

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

The radius of convergence, usually written as R, is the radius of the largest open disc around the center (here typically z_0 = 0) on which this series converges absolutely. In other words, the series converges for all complex numbers z with |z| < R, and it diverges for all |z| > R. On the boundary |z| = R, the behavior can be mixed: some points may give convergence and others divergence.

Understanding the radius of convergence is essential when you work with Taylor or Maclaurin series, solve differential equations using power series, or approximate analytic functions numerically.

Root test and ratio test formulas

There are two standard tools for finding the radius of convergence of a power series: the root test and the ratio test.

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).

Textbooks commonly ask, “How do you find the radius of convergence of a power series?” and then demonstrate either the root test or the ratio test, depending on the structure of the coefficients.

How this calculator estimates R

In practice, you rarely have access to all infinitely many coefficients a_n. Instead, you may know only a finite list, either from a symbolic computation or a numerical procedure. This calculator takes that finite list and uses a numerical version of the root test to approximate the limit superior in the formula above.

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 settle into a regular pattern (for example, factorial growth or a simple geometric factor).

How to use the calculator

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 up to which the series converges absolutely. If the estimated R is large, the series converges on a wide disc; if it is small, the disc of convergence is narrow.

Interpreting the radius of convergence

Once you have an estimate for R, it helps to connect it to several related ideas:

• Disc of convergence: The series converges absolutely for all complex numbers with |z| < R.
• Interval of convergence (real variable): If you restrict to real x, the series converges for |x| < R. On the interval endpoints x = \pm R, you must test convergence separately. The set of real x for which the series converges is called the interval of convergence, while R is just its radius when measured from 0.
• Singularities: For analytic functions, the radius of convergence equals the distance from the center to the nearest singularity in the complex plane. The calculator therefore gives insight into where the function can be analytically continued.

Worked examples

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 list of coefficients.

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

Limitations and assumptions

This tool is designed as an educational and exploratory aid rather than a formal proof engine. Its estimate of the radius of convergence relies on several important assumptions and has some limitations you should keep in mind:

• Finite data: The method uses only the coefficients you provide. 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 (for example, the series starts at a higher power of z), the tool skips undefined roots (like 0^{1/0}) and works with indices where k \ge 1. This is usually harmless, but you should be aware of how the indexing aligns with your symbolic formula.
• 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 simply 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 consider applying the exact root or ratio tests by hand as a comparison.

Who this calculator is for

This radius of convergence calculator is intended for students in calculus, real analysis, or complex analysis courses, as well as practitioners who encounter power series in applied work. It helps you:

• 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 (root test, ratio test, or knowledge of singularities), it provides quick, visual feedback on the convergence behavior of your power series.