Euler’s Number from the Factorial Series
Approximate e by summing the first requested terms of Σ1/k!, beginning with k=0, using compensated floating-point addition and a 10,000-term resource limit.
Description
Euler’s number e is the sum of reciprocal factorials starting with 1/0!.
Euler’s number e is the sum of reciprocal factorials starting with 1/0!. A finite prefix gives an approximation with rapidly shrinking positive terms.1
The tool returns value. The indexing and output conventions below determine how to read these values.
When to use Euler’s Number from the Factorial Series
Use this for the following task: Approximate e by summing the first requested terms of Σ1/k!, beginning with k=0, using compensated floating-point addition and a 10,000-term resource limit. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Euler’s Number from the Factorial Series and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter terms according to the field table, then run the tool.
- Read value. Use the stated units, indexing and interpretation of the numerical result.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| terms | Number of terms (resource limit 10000) | integer; minimum 1; maximum 10000 | Required |
Initialize the first term and sum at one. Generate later terms by dividing by k and add them with compensation, stopping if the term underflows to zero.
e approximation = Σ[k=0..terms−1]1/k! The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| value | Factorial series approximation | number |
{
"input": {
"terms": 4
},
"output": {
"value": 2.6666666666666665
}
}Limitations and assumptions
Accept 1..10000 terms. The result uses floating-point arithmetic and returns no certified error bound; terms counts include the initial k=0 term.
Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.
Alternative or Complementary methods
Choose Pi from the Leibniz Series when you need its operation: Approximate π with the requested terms of 4Σ(−1)^k/(2k+1), grouping adjacent terms to reduce cancellation; computation is capped at 100,000 terms. Choose Arithmetic–Geometric Mean Iterates for the complementary operation: Approximate the arithmetic–geometric mean of two finite nonnegative values with the requested number of iterations, returning both final iterates and limiting work to 128 iterations.
Background reference: Digital Library of Mathematical Functions, §4.2.
References
-
Digital Library of Mathematical Functions, §4.2 — National Institute of Standards and Technology
Similar or alternative tools
- Pi from the Leibniz Series
Approximate π with the requested terms of 4Σ(−1)^k/(2k+1), grouping adjacent terms to reduce cancellation; computation is capped at 100,000 terms.
- Arithmetic–Geometric Mean Iterates
Approximate the arithmetic–geometric mean of two finite nonnegative values with the requested number of iterations, returning both final iterates and limiting work to 128 iterations.