Interval of Convergence Calculator
Find the radius and interval of convergence for any power series using the ratio test. Get step-by-step solutions with endpoint tests. Choose from common series examples or enter your own coefficients.
Power Series
Center of Series (c)
The center c defines the point around which the series is expanded. For Σ aₙ(x − c)ⁿ, if your series uses x with no shift, c = 0.
Series Coefficients (first 10 terms)
a0
1.0000
a1
1.0000
a2
1.0000
a3
1.0000
a4
1.0000
a5
1.0000
a6
1.0000
a7
1.0000
a8
1.0000
a9
1.0000
Classic geometric series. Ratio |aₙ/aₙ₊₁| = 1/1 = 1 for all n, so R = 1. At x = ±1, the series is Σ (±1)ⁿ which diverges.
Interval of Convergence
Interval
(-1.0000, 1.0000)
Apply the Ratio Test
For power series Σ aₙ(x − 0)ⁿ, the radius R = lim n→∞ |aₙ / aₙ₊₁|.
Compute consecutive ratios |aₙ/aₙ₊₁|
n=0: 1.0000, n=1: 1.0000, n=2: 1.0000, n=3: 1.0000, n=4: 1.0000, n=5: 1.0000, ... (9 ratios total)
Radius of Convergence
R ≈ 1
Open interval (before endpoint tests)
(-1.0000, 1.0000)
Test left endpoint x = c − R = -1.0000
Substituting x = -1.0000 into the series and examining partial sums.
Test right endpoint x = c + R = 1.0000
Substituting x = 1.0000 into the series and examining partial sums.
Final Interval of Convergence
(-1.0000, 1.0000)
Endpoint Tests
Partial sums do not stabilize - series likely diverges at this endpoint.
Endpoint behavior is inconclusive with the given terms. More terms or a symbolic test is needed.
Common Power Series Intervals of Convergence
These are the standard series you will encounter in a Calculus 2 course.
| Series | Type | Radius R | Interval | Notes |
|---|---|---|---|---|
| Σ xⁿ | Geometric | 1 | (−1, 1) | Both endpoints excluded - series diverges at ±1 |
| Σ xⁿ/n | Harmonic power | 1 | [−1, 1) | x = −1 converges (alternating harmonic), x = 1 diverges |
| Σ xⁿ/n² | p-series power | 1 | [−1, 1] | Both endpoints included - p-series with p=2 converges |
| Σ xⁿ/n! | Exponential (eˣ) | ∞ | (−∞, +∞) | Converges for all real x |
| Σ n·xⁿ | n coefficient | 1 | (−1, 1) | Endpoints excluded - terms don't approach zero at ±1 |
How to Use the Calculator
Select a Series or Enter Custom Coefficients
Choose from five preset examples (geometric, harmonic, exponential, alternating, or n² series) to see worked solutions instantly. For your own series, select 'Custom Coefficients' and enter the aₙ values as comma-separated numbers. Include at least 8-10 terms for a stable radius estimate.
Set the Center (c)
Enter the center of your power series. For most textbook problems this is 0, but any real number works. The center shifts the interval symmetrically: if c = 3 and R = 2, the interval is (1, 5).
Read the Interval and Steps
The results panel shows the radius R and the final interval with correct bracket notation. The step-by-step solution walks through the ratio test, radius computation, and both endpoint tests. The ratio table shows the consecutive ratios converging toward R.
Check Endpoint Results
Each endpoint test shows whether the series converges or diverges at that boundary point, with a note about the partial sum behavior. This tells you whether to use [ ] or ( ) for each side of the interval.
Calculator Features
Ratio Test Implementation
Computes |aₙ/aₙ₊₁| for each pair of consecutive coefficients and estimates R from the limit of those ratios.
Five Preset Examples
Geometric, harmonic, exponential (eˣ), alternating harmonic, and n² series - all with known correct answers you can verify against.
Endpoint Convergence Testing
Numerically tests each endpoint by computing partial sums at x = c±R to determine if they stabilize (converge) or diverge.
Step-by-Step Solution
Shows each step of the ratio test process - coefficient ratios, radius computation, and endpoint test conclusions.
Ratio Values Table
Displays the consecutive ratios |aₙ/aₙ₊₁| so you can see how they converge toward R - helpful for understanding the limit process.
Custom Coefficient Input
Enter any numeric coefficients as comma-separated values including fractions like 1/3. Supports series from any calculus textbook.
Understanding Interval of Convergence
What Is a Power Series?
A power series is an infinite sum of the form Σ aₙ(x − c)ⁿ, where aₙ are coefficients, c is the center, and x is the variable. You can think of it as a polynomial with infinitely many terms. The key question is: for which values of x does this infinite sum converge to a finite number instead of blowing up to infinity?
The interval of convergence answers this question exactly. It tells you the range of x values where the series behaves well - where you can substitute x and get a meaningful finite answer. Understanding this interval is fundamental to working with Taylor and Maclaurin series, which are used everywhere in calculus, physics, and engineering to approximate functions that would otherwise be impossible to compute directly.
The Ratio Test: The Standard Approach
The ratio test is the go-to method for finding the interval of convergence because it works directly with the coefficients. You form the ratio |aₙ₊₁/aₙ|, multiply by |x − c|, and take the limit as n approaches infinity. The series converges wherever this limit is less than 1.
Solving L × |x − c| < 1 (where L = lim |aₙ₊₁/aₙ|) gives you |x − c| < 1/L = R. This is the key insight: R = 1/L = lim |aₙ/aₙ₊₁|. If L = 0, then R = ∞ and the series converges everywhere. If L = ∞, then R = 0 and the series only converges at x = c.
Why Endpoint Testing Cannot Be Skipped
At x = c ± R, the ratio test gives L = 1 exactly - the test breaks down and tells you nothing. The series might converge or diverge at these boundary points, and you have to check each one manually. Different series behave differently at their endpoints even when they have the same radius R.
The harmonic series family illustrates this perfectly. Σ xⁿ/n and Σ xⁿ/n² both have R = 1. But Σ xⁿ/n has interval [−1, 1) because it converges at x = −1 (alternating harmonic) and diverges at x = 1 (harmonic series). Meanwhile Σ xⁿ/n² has interval [−1, 1] because at both endpoints you get a p-series with p = 2 > 1, which converges. The radius alone cannot tell you which case you are in.
Real-World Applications of Convergence
Power series and their intervals of convergence are not just abstract math exercises - they underpin how computers compute trigonometric functions, logarithms, and exponentials. Your calculator finds sin(0.5) not by any direct formula but by summing the Taylor series Σ (−1)ⁿx²ⁿ⁺¹/(2n+1)! which converges for all x. The fact that this series has an infinite radius of convergence means the computation works for any input.
In physics, series with finite convergence intervals appear in quantum mechanics, heat transfer equations, and electric field calculations. Engineers use power series approximations to linearize nonlinear systems near operating points - a technique that only works within the series' interval of convergence. Understanding where a series converges is the difference between a valid approximation and a completely wrong answer.
Interval of Convergence Calculator FAQ
Common questions about power series, the ratio test, and convergence intervals.