CALCULATE WITH CONFIDENCE

Sequences & Series

Power Series Convergence Calculator

Find the radius and open interval of absolute convergence from a coefficient-ratio limit.

Sequences & Series

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  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example

Enter finite decimal or scientific-notation numbers, and comma-separated terms where requested. Inputs stay on this device.

Result

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Understand the subject

What is Power Series Convergence?

A sequence follows an ordered rule; a series adds selected sequence terms into one total.

Use it to generate, predict, add, or test an ordered pattern while keeping its rule explicit.

The relationship

Write the rule before calculating

See the structure

What the calculation is doing

Worked interpretation

Read the result in context

Center 2 and L=1/3 give radius 3 and open interval (−1,5).

Interpret with care

Important boundary

A few initial terms can fit many possible rules; a generated pattern is not proof of a unique universal formula.

Use the Mathematics collection to move between connected concepts without duplicating the calculation.

Quick guide

How to use this calculator

  1. Choose the supported sequence, series, or analysis mode when the page offers one.
  2. Enter the labelled parameters or comma-separated finite terms.
  3. Read the primary result, then use the supporting classification, formula, or generated values.

Calculation method

Apply the stated sequence or series rule

If L=lim|aₙ₊₁/aₙ|, then R=1/L.

The calculator validates the selected family and its mathematical domain before returning a result.

Worked example

Worked example

Center 2 and L=1/3 give radius 3 and open interval (−1,5).

If L=lim|aₙ₊₁/aₙ|, then R=1/L.

Supported inputs

Precision and limits

Supported scope

General-looking pages use explicitly listed families instead of pretending to parse or prove arbitrary symbolic sequences and series.

Numerical precision

Fibonacci, Lucas, Tribonacci, Pascal, and triangular-number outputs use exact integers. Analytical and decimal calculations use finite double-precision arithmetic and normally display 12 significant digits.

Visible computational limits

Named-sequence indices are limited to 0–5000; Pascal rows to 0–200; harmonic indices to 1–1,000,000; recurrence and arithmetic-geometric generation to 1000; Taylor degree to 100; Lagrange degree to 170; sigma bounds to −100,000…100,000 with at most 200,000 steps; pattern and difference lists to 100 and 50 terms. Field labels show narrower limits where applicable.

Convergence

A convergence test applies only when its hypotheses match the entered series. Inconclusive tests and power-series endpoints are reported honestly rather than guessed.

Calculator-specific rule

Finite endpoints require separate tests and are not inferred from L.