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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 says so instead of showing 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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