Calculator Input
Example Data Table
| Function or Coefficient | an | Expected Radius | Endpoint Note |
|---|---|---|---|
| ex | 1 / n! | ∞ | Converges for every real x. |
| Geometric series | qn | 1 / |q| | Endpoints usually diverge. |
| Harmonic power series | 1 / (n + 1) | 1 | Left endpoint converges conditionally. |
| Quadratic decay | 1 / (n + 1)2 | 1 | Both endpoints converge. |
Formula Used
A Maclaurin series has the form
Σ a_n x^n. The radius of convergence is commonly found with the ratio test:
R = lim |a_n / a_{n+1}|
The root test can also be used:
R = 1 / limsup |a_n|^(1/n)
The open interval is (-R, R). Each endpoint must be tested separately.
Endpoint behavior may depend on p-series, alternating series, or term tests.
How to Use This Calculator
Select a coefficient model. Enter p, q, a custom formula, or a coefficient list when needed. Choose the number of sample terms. Add a test value for x. Press the calculate button. The result appears below the header and above the form. Review the radius, interval, endpoint status, graph, and table. Use CSV or PDF buttons to save your result.
Understanding Maclaurin Radius of Convergence
What the Radius Means
A Maclaurin series expands a function around zero. It writes the function as a power series. The radius of convergence tells where that power series behaves well. Inside the radius, the terms usually shrink and the series converges. Outside the radius, the terms normally fail to shrink fast enough. The center is zero, so the basic interval begins as a symmetric range around zero.
Why Coefficients Matter
Every coefficient sequence creates a different convergence pattern. Factorial denominators often create very wide convergence. Factorial growth creates a very narrow result. Polynomial decay often gives radius one. Geometric coefficients give a radius controlled by the multiplier. This calculator compares these patterns and estimates the radius from sample terms.
Ratio and Root Tests
The ratio test studies the size of one coefficient compared with the next coefficient. It is fast and useful for many common sequences. The root test studies the nth root of each coefficient. It can work better when many coefficients are zero or irregular. The calculator shows both estimates, so the result is easier to inspect.
Endpoint Testing
The radius gives only the open interval first. The endpoints need separate checks. A series may converge at both endpoints, one endpoint, or neither endpoint. For example, a p-series endpoint may require p greater than one. An alternating endpoint may converge conditionally when the positive version fails. This is why endpoint notes are shown beside the interval.
Practical Use
Use this tool while studying Taylor series, differential equations, complex analysis, and numerical approximation. The graph helps reveal partial sum behavior. The table shows coefficients, terms, ratios, roots, and partial sums. Exports help you save classroom work, assignments, or research checks.
FAQs
1. What is a Maclaurin series?
A Maclaurin series is a power series centered at zero. It represents a function using coefficients multiplied by powers of x.
2. What is radius of convergence?
It is the distance from zero where the series converges. Values inside the radius usually converge, while outside values usually diverge.
3. Why are endpoints tested separately?
The ratio or root test often gives the open interval only. Endpoint convergence depends on the actual series at each boundary.
4. What does infinite radius mean?
Infinite radius means the Maclaurin series converges for every real x. Series for e^x, sin(x), and cos(x) have this behavior.
5. What does radius zero mean?
Radius zero means the series only converges at the center, usually x = 0. Factorial coefficient growth can cause this result.
6. Which test is more reliable?
The ratio test is simple for regular coefficients. The root test can be better for irregular or sparse coefficient patterns.
7. Can I enter my own coefficient formula?
Yes. Choose the custom formula option. Use n as the variable. Common functions like pow, sin, cos, exp, and log are supported.
8. Why is my custom result approximate?
Custom formulas are estimated from finite samples. For difficult series, confirm endpoints and limits with symbolic calculus methods.