Interval of convergence calculator

A power series Σ aₙ(x − c)ⁿ only converges to a finite value for certain x. This calculator estimates the radius of convergence R using the ratio test, then gives the interval of convergence centred on c. Enter the coefficient aₙ as a function of n (for example 1/n, 2^n, or n!) and the centre c.

How the ratio test works

The radius of convergence is R = limₙ→∞ |aₙ / aₙ₊₁|. The series converges when |x − c| is less than R and diverges when it is greater, giving the open interval (c − R, c + R). The calculator evaluates the ratio for large n to estimate R; if the ratio grows without bound the radius is infinite, and if it shrinks to zero the series converges only at its centre.

Don't forget the endpoints

The ratio test is always inconclusive at the two endpoints, so each must be checked separately by substituting it into the series and testing that series for convergence. The calculator flags the endpoints for you. For related work, see our calculus solver.

Frequently asked questions

How do I enter a factorial?

Type n! (for example in 1/n!). The calculator understands factorial notation.

Is the result exact?

It is a reliable numerical estimate of the radius. For an exact answer, apply the ratio test algebraically and use this as a check.