Abel Sum Calculator
Compute finite radial Abel-weighted series Σ a_n r^n for supplied coefficients; no unprovided tail or r→1 limiting summability claim.
Description
Abel weighting applies a radial factor r^n to a sequence beginning at index zero.
Abel weighting applies a radial factor r^n to a sequence beginning at index zero. Evaluating a finite weighted prefix is different from taking an infinite r→1 limit.1
Compute finite radial Abel-weighted series Σ a_n r^n for supplied coefficients; no unprovided tail or r→1 limiting summability claim.
When to use Abel Sum Calculator
Use this tool to compute finite radial Abel-weighted series Σ a_n r^n for supplied coefficients; no unprovided tail or r→1 limiting summability claim.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Abel Sum Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter seriesTerms, radialWeight using the field table and calculation rules, then run the tool.
- Read finiteAbelWeightedSum, lastIncludedWeightedTerm. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| seriesTerms | Series terms | list of number | Required |
| radialWeight | Radial weight | number | Required |
Enter 1–10,000 finite coefficients and r in [0,1). Multiply successive terms by powers of r, starting at power 1, and return the sum and last included weighted term. At r=0 only the index-zero contribution survives.
finite weighted sum = Σ[n=0..N−1]a_n r^n Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| finiteAbelWeightedSum | Finite abel weighted sum | number |
| lastIncludedWeightedTerm | Last included weighted term | number |
{
"input": {
"seriesTerms": [
1,
-1,
1,
-1
],
"radialWeight": 0.5
},
"output": {
"finiteAbelWeightedSum": 0.625,
"lastIncludedWeightedTerm": -0.125
}
}Limitations and assumptions
No unprovided tail or Abel-limit summability claim is supplied.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Cesàro Sum Calculator: Compute finite Cesàro mean of successive partial sums from actual series terms, not mean of raw terms or claimed convergence/infinite sum. For a complementary task, use Euler Transform Calculator: Compute finite Euler alternating-series transform Σ Δ^k a_0/2^(k+1), using Δa_n=a_n−a_(n+1); preserves finite differences and makes no infinite-series convergence promise.
References
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Abel Sum Calculator — tool.io
Similar or alternative tools
- Cesàro Sum Calculator
Compute finite Cesàro mean of successive partial sums from actual series terms, not mean of raw terms or claimed convergence/infinite sum.
- Euler Transform Calculator
Compute finite Euler alternating-series transform Σ Δ^k a_0/2^(k+1), using Δa_n=a_n−a_(n+1); preserves finite differences and makes no infinite-series convergence promise.