Description
Calculate a signed infinite geometric tail from its first omitted term and ratio.
A sequence rule specifies how terms relate. Checking one transition, advancing a recurrence, and bounding an infinite tail answer different questions. A calculation from supplied coefficients assumes that rule; it does not infer it from observations.
Calculate a signed infinite geometric tail from its first omitted term and ratio.
When to use this tool
Use it for geometric partial sum tail ratio when the required inputs and conventions are already available.
How to use the tool and how it works
- Enter ratio, firstOmitted. Use the stated units and boundary conventions.
- Run the calculation. Correct any input error before interpreting a result.
- Read tail. Inspect any known, defined, inclusion, or boundary flags before interpreting numeric values.
Sum the omitted geometric tail as firstOmitted/(1-ratio), requiring abs(ratio) below one. The sign of the ratio remains part of the calculation.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| ratio | Dimensionless ratio of compatible quantities under the stated formula. | Required. Finite number. |
| firstOmitted | First Omitted | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| tail | Tail | Required. Finite number. |
{
"input": {
"ratio": 0.5,
"firstOmitted": 0.125
},
"output": {
"tail": 0.25
}
}Limitations and assumptions
- This is the signed infinite tail under a geometric model, not an absolute error bound. The tool does not verify that actual later terms follow the supplied ratio.
- The field tables describe structural bounds; the mechanics also state cross-field requirements. Malformed input shapes, disallowed keys, and invalid domains cause input errors. Results must remain finite and within the declared output contract. A successful false flag or zero placeholder is distinct from an error.
- Numeric arithmetic uses binary floating point. Exact comparisons have no hidden tolerance or decimal rounding; negative zero is exported as zero. Decimal-text integer operations retain the exactness specifically stated above.
Alternative or Complementary methods
For geometric partial sum tail ratio, check the source records and the supplied rule before comparing the result with another method. Use exact integer or symbolic arithmetic when a numerical equality is insufficient.
References
Similar or alternative tools
- Geometric Series Single Tail Bound
Calculate an absolute tail bound from a first omitted magnitude and a supplied geometric domination ratio.
- Arithmetic Geometric Single Term Formula
Evaluate one zero-based term of a supplied arithmetic-geometric progression.