Calculator Inputs
Example Data Table
These examples show common endpoint behavior for different series types.
| Series | Center | Parameter | Radius | Endpoint Result | Interval |
|---|---|---|---|---|---|
| Σ (x − 0)ⁿ / n² | 0 | p = 2 | 1 | Both endpoints converge absolutely | [-1, 1] |
| Σ (x − 3)ⁿ / n | 3 | p = 1 | 1 | Left conditional, right divergent | [2, 4) |
| Σ 2ⁿ(x + 1)ⁿ | -1 | k = 2 | 0.5 | Both endpoints diverge | (-1.5, -0.5) |
| Σ (x − 5)ⁿ / n! | 5 | Factorial denominator | ∞ | No finite endpoint needed | (-∞, ∞) |
Formula Used
General power series:
Σ aₙ(x − c)ⁿ
The calculator uses the ratio-test form:
L = lim |aₙ₊₁ / aₙ|.
When this limit exists, the radius is:
R = 1 / L
If L = 0, the radius is infinite. If the ratio grows without bound,
the radius is zero. After finding the radius, the open interval
(c − R, c + R) is tested at both endpoints.
Endpoint tests may use p-series rules, alternating-series rules, absolute convergence, or divergence by the nth-term test.
How to Use This Calculator
- Select the coefficient pattern that matches your power series.
- Enter the center value c from the expression x − c.
- Enter p, k, or the custom ratio limit when needed.
- For custom mode, choose endpoint test results if already known.
- Press Calculate Interval to view the radius and interval.
- Use CSV or PDF download buttons to save the report.
Understanding Interval of Convergence
What the Interval Means
An interval of convergence tells where a power series works. A power series is built around a center point. It may converge near that point and fail farther away. The interval gives the exact x-values where the infinite sum has a finite value.
Why Radius Comes First
Most interval problems start with a radius. The radius measures the distance from the center to the boundary of convergence. The ratio test is the most common method. It compares neighboring terms. When the comparison is less than one, the series converges. This condition usually becomes an inequality involving x.
Why Endpoints Need Extra Work
The ratio test normally gives an open interval. It does not settle the boundary points.
Each endpoint must be substituted back into the original series. One endpoint may
converge while the other diverges. This is why intervals may use mixed brackets, such as
[a, b) or (a, b].
Common Endpoint Patterns
Series with 1 / nᵖ often use p-series rules. The right endpoint becomes a
positive p-series. The left endpoint often becomes alternating. Alternating series can
converge conditionally when the positive version diverges. Factorials create stronger
behavior. A factorial in the denominator often gives convergence for every real x.
A factorial in the numerator usually leaves only the center point.
Practical Use
This tool is useful for calculus homework, quick checking, and study notes. It does not replace written reasoning. Instead, it organizes the key steps. You can compare the radius, endpoints, formulas, and final interval. The export options also help create a clean record for review.
FAQs
1. What is an interval of convergence?
It is the set of x-values where a power series converges. It may be open, closed, half-open, infinite, or only one point.
2. What is the radius of convergence?
The radius is the distance from the center to the boundary where convergence can change. It creates the main open interval.
3. Why are endpoints tested separately?
The ratio test usually gives only a strict inequality. Endpoints make the ratio equal one, so another test is required.
4. What does conditional convergence mean?
Conditional convergence means the series converges, but its absolute-value series diverges. Alternating endpoint series often behave this way.
5. What happens when the radius is infinite?
The series converges for every real x. The interval is written as negative infinity to positive infinity.
6. What happens when the radius is zero?
The series converges only at its center. For all other x-values, the terms usually fail the convergence condition.
7. Can this calculator handle custom coefficients?
Yes. Use custom ratio mode when you already know the limit of absolute neighboring coefficient ratios.
8. Does the calculator prove every endpoint?
It proves supported patterns automatically. For custom coefficients, you should enter endpoint test results from your own separate analysis.