What Is the Interval of Convergence?

A power series, something like ∑ aₙ(x − c)ⁿ from n=0 to ∞, looks like an infinite polynomial. For a given x, that sum is either a finite number or it is not. The set of x where it is a finite number is the interval of convergence. Without that interval, the series is a formula for nothing; with it, you can use the series to represent functions, differentiate term by term, or approximate values. The question of what is interval of convergence is answered by one definition: it is the complete set of x for which a power series converges to a finite sum.

A power series is an infinite sum ∑ cₙ(x − a)ⁿ, where a is the center and cₙ are constants. Every power series has a radius of convergence R that says how far from the center you can go before the series might diverge. Inside that radius, on the open interval (a − R, a + R), the series converges absolutely. Outside it, the series diverges. The ratio test and root test, as taught in Stewart's Calculus (sections 11.6-11.8), are the standard tools for finding R. But R alone never tells you what happens at the endpoints x = a ± R; those must be checked separately.

The interval of convergence is the actual set of x for which the power series converges, endpoints included or excluded as the series dictates. It is always centered at a, and its half-length is the radius R. The open interval (a − R, a + R) is guaranteed convergence. To get the full interval, test each endpoint by plugging it into the original series and applying a convergence test, p-series, alternating series test, or direct comparison, because the ratio test is inconclusive at L = 1. OpenStax Calculus Vol. 2 (sections 5.5-5.6) does exactly this: find R, then test endpoints.

A power series can converge in exactly three ways. First, it may converge only at its center. When R = 0, the interval is {a}, a single point. This happens when coefficients grow too fast, like aₙ = n!. Second, it may converge for all real x. When R = ∞, the interval is (−∞, ∞). The Maclaurin series for eˣ, sin x, and cos x all have R = ∞. Third, and most common, it converges on an interval of finite length 2R. That interval may be open, half-open, or closed. For example, the series for 1/(1 − x) converges on (−1, 1) (open), while the series for ln(1 + x) converges on (−1, 1] (closed on the right). The difference is always endpoint behavior.

A power series is a representation of a function only inside its interval of convergence. Outside that interval, the series diverges, it doesn't equal the function, and it doesn't equal anything finite. The classic example is 1/(1 − x) = ∑ xⁿ. The function is defined for all x ≠ 1, but the series converges only for |x| < 1. At x = 2, the function equals −1, yet the series 1 + 2 + 4 + 8 + … diverges to infinity. The series representation is valid only on its interval of convergence. Trying to use the series outside that interval produces false results. This is why Stewart (section 11.10) and OpenStax (section 6.3) always state the interval alongside the Taylor series of a function.

The interval of convergence definition is simple: the set of x where the series works. In practice, newcomers confuse the radius (a number) with the interval (a set). The radius tells you how far from the center you can go; the interval tells you exactly which x are safe. A student who finds R = 2 and writes the interval as (−2, 2) has not finished, they must test x = −2 and x = 2. The series might converge at one, both, or neither. That extra step is where most mistakes happen. OpenStax (section 5.5) warns that the ratio test is inconclusive at endpoints, yet students still skip the check.

Power series convergence depends entirely on the value of x relative to the center. For a given series ∑ cₙ(x − a)ⁿ, the ratio test gives a limit L that is a function of x. When L < 1, the series converges; when L > 1, it diverges. Setting L < 1 yields |x − a| < R, which is the open interval. The root test is an alternative that works when the ratio test limit is messy, especially for terms with exponents or factorials. Both tests come from Stewart (sections 11.6-11.7) and are the primary methods for finding R. But neither can tell you about endpoint behavior, because at endpoints the limit equals 1.

One common failure: assuming that if x = a + R converges, then x = a − R must also converge. The harmonic series at one endpoint and an alternating harmonic at the other is the classic counterexample. OpenStax (section 5.5) demonstrates this with the series ∑ xⁿ/n, which converges at x = −1 but diverges at x = 1. Always test each endpoint separately.

The interval of convergence meaning is the answer to: "For which x does this infinite sum make sense?" The radius tells you the distance; the interval tells you the actual numbers. For the series ∑ (x − 2)ⁿ/(n³ + 1), the radius is 1, but the interval is [1, 3] because both endpoints converge. For ∑ n! xⁿ, the radius is 0 and the interval is just {0}. The interval is always a subset of the real line, symmetric in length about the center, though the endpoint inclusion can be asymmetric. This precision matters in applications: an engineer using a Taylor series to approximate a function must stay inside the interval, or the approximation is invalid.

A graph of partial sums against the original function makes the interval visible. Take f(x) = 1/(1 − x) and its power series ∑ xⁿ. Plot the function and the partial sums S₁(x) = 1 + x, S₂(x) = 1 + x + x², S₃(x) = 1 + x + x² + x³, up to S₅(x), on the interval (−2, 2). Inside (−1, 1), each partial sum gets closer to the curve as n increases. At x = 0.5, S₅ differs from f(x) by about 0.03. At x = 1.2, outside the interval, S₅ is about 7.8 while f(1.2) = −5. The partial sums diverge away from the function. The graph shows exactly where the series representation breaks down. Stewart (section 11.10) includes such graphs to illustrate the convergence behaviour.

The single thing that most often goes wrong: students find the radius, write the open interval, and stop. They forget to test endpoints. That one missing step turns a correct R into an incorrect interval. Always test both endpoints separately, and do not assume symmetry.