About This Radius Tool
A power series usually looks simple, yet its useful range can be delicate. This calculator helps estimate the radius of convergence for series written around a center. It supports ratio test limits, root test limits, and common coefficient templates. You can also record endpoint behavior, so the final interval note is more practical.
Why Radius Matters
The radius tells how far x may move from the center before the series stops converging. Inside that distance, a power series behaves like a trusted function. Outside it, terms normally fail to settle. At the boundary, separate checks are needed, because the radius test gives no final answer there.
Advanced Input Choices
Use the ratio limit when you know the limit of the absolute coefficient ratio. Use the root limit when the nth root of the coefficient is easier. For common templates, choose geometric, polynomial geometric, factorial denominator, or factorial numerator. These options cover many textbook patterns. The calculator also accepts a center value, so results can be shown as an interval around that point.
Endpoint Review
Endpoint testing is optional because it depends on the original series. A finite radius gives two boundary points. You may enter notes for the left endpoint and right endpoint. For example, an alternating harmonic endpoint may converge, while a harmonic endpoint may diverge. The tool preserves these notes in the result and export files.
Learning Benefit
The calculator is meant to show reasoning, not only an answer. It displays the selected method, the limit used, the radius, and the interval form. This makes it useful for checking homework, preparing examples, or building content for calculus lessons. The example table gives sample inputs and expected outcomes.
Good Practice
Always verify that your coefficient model matches the series. If the coefficient contains factorials, powers, or exponentials, choose the closest template. If you already computed a limit by hand, use ratio or root mode. For endpoint decisions, test the two substituted series separately. This keeps the final interval accurate and clearly justified.
Best Results
Use decimal values when a limit is numeric. Use infinity only when the coefficients shrink faster than every exponential rate. Use zero when coefficients grow too quickly for any nonzero distance.