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
G(x)=Σₙ₌₀∞aₙxⁿ
See the structure
What the calculation is doing
a₁rulea₂rulea₃G(x)=Σₙ₌₀∞aₙxⁿ
Worked interpretation
Read the result in context
The Fibonacci sequence has G(x)=x/(1−x−x²).
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
Choose the supported sequence, series, or analysis mode when the page offers one.
Enter the labelled parameters or comma-separated finite terms.
Read the primary result, then use the supporting classification, formula, or generated values.
Calculation method
Apply the stated sequence or series rule
G(x)=Σₙ₌₀∞aₙxⁿ
The calculator validates the selected family and its mathematical domain before returning a result.
Worked example
Worked example
The Fibonacci sequence has G(x)=x/(1−x−x²).
G(x)=Σₙ₌₀∞aₙxⁿ
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
This page returns formal ordinary generating functions for the listed families; it does not parse arbitrary recurrences.
