Interval of Convergence Formulas: Ratio Test and Root Test

Understanding the Interval of Convergence Formula

The interval of convergence of a power series tells you for which x values the series adds up to a finite number. The key to finding this interval is the ratio test or the root test. Both tests use a simple formula to compute the radius of convergence, R. Once you have R, the interval is centered at the series center c and stretches from c - R to c + R. But we must also check the endpoints separately. This page breaks down the formula step by step so you can understand what each symbol means and why it works.

The Power Series and the Ratio Test Formula

Every power series looks like this:

Σ aₙ (x - c)ⁿ = a₀ + a₁(x - c) + a₂(x - c)² + a₃(x - c)³ + …

Here aₙ is the coefficient of the nth term and c is the center point. The series converges for x values close to c. The ratio test formula used by our Interval of Convergence Calculator is:

L = lim (n → ∞) |aₙ₊₁ / aₙ|

Then the radius of convergence is:

R = 1 / L

Let’s break it down:

• aₙ – the coefficient of the nth term.
• aₙ₊₁ – the coefficient of the next term (replace n with n+1).
• |aₙ₊₁ / aₙ| – the absolute ratio of consecutive coefficients.
• L – the limit of that ratio as n goes to infinity.

The ratio test works because it compares how fast the terms shrink. If the ratio of successive terms approaches a number L, then the series behaves like a geometric series with ratio L|x-c|. For convergence we need L|x-c| < 1, which gives |x-c| < 1/L = R. That’s why R is the reciprocal of L.

The Root Test: Another Way to Find R

The root test is useful when the aₙ terms involve powers or factorials. Its formula is:

L = lim (n → ∞) √[n]{|aₙ|}

where √[n]{|aₙ|} means the nth root of the absolute value of aₙ. Again,

R = 1 / L

The root test looks at the overall size of the terms after taking the nth root. It gives the same radius as the ratio test when both apply, but sometimes it is easier to compute. For example, when aₙ = (nⁿ), the root test simplifies nicely. Our calculator automatically picks the best test, but you can also choose one manually.

From Radius to Interval: Interpreting the Results

Once you have R from either formula, the interval of convergence is centered at c with half-width R. That gives the open interval (c - R, c + R). However, the series might also converge at one or both endpoints. You must plug x = c - R and x = c + R into the series and test separately using another convergence test (like the alternating series test or p-series test).

For a deeper explanation of these steps, check out our page on What Is Interval of Convergence? Definition & Examples (2026) and the step-by-step guide on How to Calculate Interval of Convergence (2026).

Edge Cases: Special Forms and What They Mean

• R = 0: The series converges only at x = c. This happens when L → ∞ in the ratio or root test.
• R = ∞: The series converges for all x (the interval is (-∞, ∞)). This occurs when L = 0.
• Taylor and Maclaurin series: These are special power series where c = 0 (Maclaurin) or some other center. Their interval of convergence follows the same formulas. See our page on Interval of Convergence for Taylor and Maclaurin Series (2026) for more.
• Factorial and exponential coefficients: Often lead to R = ∞ because factorial grows faster than any power, making L = 0.
• Alternating signs: The ratio test handles that automatically because we take absolute values. The sign does not affect the radius, but it can affect endpoint convergence.

Historical Origins and Practical Applications

The ratio test was first published by Jean le Rond d'Alembert in 1768, and the root test was later refined by Augustin-Louis Cauchy. Both tests are based on comparing a series to a geometric series, a method used since the 17th century. Today, these formulas are essential in calculus for expanding functions into power series, solving differential equations, and analyzing numerical methods. Engineers and scientists use them to approximate functions with polynomials.

Putting It All Together

To recap, the interval of convergence formula is simple: compute L using either the ratio or root test, then R = 1/L. The interval is (c - R, c + R) plus possibly endpoints. If you have a power series and need a quick, accurate result, try our Interval of Convergence Calculator. It does the heavy lifting and shows step-by-step solutions.