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Interval of Convergence Calculator

Enter the center and radius of convergence.

How it is calculated

For a power series Σ cₙ (x − a)ⁿ you give the center a and the radius of convergence R, which you find beforehand, usually with the ratio test: R = lim |cₙ / cₙ₊₁|. The tool returns the open interval (a − R, a + R) with four decimals, plus the left and right ends separately. The radius must be zero or positive; a negative radius gives no result. If R = 0 the series converges only at the center, and the tool shows that single point, x = a, instead of an interval. It has no way to enter an infinite radius. It never tests the two endpoints, so the interval is always shown open: whether the series converges at a − R and a + R needs a separate check.

Worked example

Consider the series Σ (x − 2)ⁿ / 3ⁿ. The ratio test gives R = 3 and the center is a = 2. Left end: 2 − 3 = −1. Right end: 2 + 3 = 5. The tool shows (-1.0000, 5.0000). For x strictly between −1 and 5 the series converges. At the endpoints you must test by hand: at x = 5 every term equals 1, and at x = −1 the terms alternate between 1 and −1. In both cases the terms do not go to zero, so the series diverges there and the open interval is the complete answer.

Try this example →

Common mistakes

  • Entering the endpoint instead of the radius. The second field is the distance R from the center, not the right end of the interval.
  • Assuming the endpoints are included because the tool prints a closed-looking pair of numbers. The interval is open; each endpoint needs its own test, such as the alternating series test or a p-series comparison.
  • Forgetting that the center is the number inside (x − a). For a series in (x + 3)ⁿ the center is −3, not 3.
  • Entering the radius of a series in x² as if it were the radius in x. For Σ x²ⁿ / 3ⁿ the ratio of coefficients gives 3, but convergence needs x² < 3, so the radius in x is √3 ≈ 1.7321.

About this calculator

A power series converges only within a certain distance of its center, and that distance is the radius of convergence R, typically found by applying the ratio test to the series' terms — taking the limit of the ratio of consecutive coefficients as the index grows. Once R is known, the series is guaranteed to converge for every x strictly between a minus R and a plus R, and to diverge outside that range.

You enter the center a and the radius R, and this calculator builds the open interval (a − R, a + R), which is the day-to-day task in calculus courses covering Taylor and Maclaurin series. If R is 0, the series converges only at the center itself. What the ratio test cannot settle is what happens exactly at the two endpoints — those require separate tests (such as the alternating series test or a p-series comparison) and are checked individually. This tool does not test the endpoints, so the interval it gives never includes them; you need to verify each endpoint on your own.

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Comments (1)

  • Editorial team

    The tool deliberately leaves the endpoints open, so finish the job by testing them yourself. For the series of x^n/n with center 0 and radius 1, the harmonic series at x = 1 diverges, while at x = -1 the alternating series converges, so the true interval is [-1, 1), not the (-1, 1) shown. Which series are you working on, and which endpoint test do you reach for?

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