Beta Dirichlet Approximation

Compute finite real Dirichlet beta at supplied s>0 with explicit odd-denominator alternating-tail bound; not a fixed Catalan lookup or invented analytic continuation.

Description

Dirichlet beta alternates over odd-denominator reciprocal powers.

Dirichlet beta alternates over odd-denominator reciprocal powers. Changing the exponent changes the function, unlike a fixed Catalan-constant lookup.^[1]

Compute finite real Dirichlet beta at supplied s>0 with explicit odd-denominator alternating-tail bound; not a fixed Catalan lookup or invented analytic continuation.

When to use Beta Dirichlet Approximation

Use this tool to compute finite real Dirichlet beta at supplied s>0 with explicit odd-denominator alternating-tail bound; not a fixed Catalan lookup or invented analytic continuation.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Beta Dirichlet Approximation and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter exponent, seriesTerms using the field table and calculation rules, then run the tool.
Read finiteDirichletBetaSum, alternatingRemainderAbsoluteBound. 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
exponent	Exponent	number	Required
seriesTerms	Series terms	number	Required

Enter positive exponent s and integer N from 1 to one million. Sum N terms and return the finite value plus the alternating-series absolute remainder bound.

beta partial sum = Σ[k=0..N−1](−1)^k/(2k+1)^s; tail bound = (2N+1)^(−s)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
finiteDirichletBetaSum	Finite dirichlet beta sum	number
alternatingRemainderAbsoluteBound	Alternating remainder absolute bound	number

json

{
  "input": {
    "exponent": 1,
    "seriesTerms": 3
  },
  "output": {
    "finiteDirichletBetaSum": 0.8666666666666667,
    "alternatingRemainderAbsoluteBound": 0.14285714285714285
  }
}

Limitations and assumptions

The tail bound excludes floating-point roundoff. Nonpositive-exponent analytic continuation is unsupported.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Dirichlet Eta Approximation: Compute finite real Dirichlet eta series for s>0 and analytic alternating-tail bound; caller exponent changes function, unlike Catalan constant fixed odd-square series. 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

Beta Dirichlet Approximation — tool.io

Similar or alternative tools

Dirichlet Eta Approximation
Compute finite real Dirichlet eta series for s>0 and analytic alternating-tail bound; caller exponent changes function, unlike Catalan constant fixed odd-square series.
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.