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Generalized Harmonic Number Calculator — result sheet
Calculate a finite generalized harmonic sum and show the last-term and average-term contributions.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: Generalized harmonic number H(n,p)=sum from k=1 to n of 1/k^p; last term=1/n^p; average term=H(n,p)/n.
A generalized harmonic number is a finite sum of reciprocal powers. Order p=1 gives the ordinary harmonic number, while other positive orders change how quickly later terms shrink. The calculator sums the requested finite sequence directly and keeps the last term visible so visitors can see the contribution that remains at the boundary.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Number of terms n — minimum 1; maximum 1000000
- Order p — minimum 1.0E-6; maximum 20
Worked example
| Input | Value |
|---|---|
| Number of terms n | 10 |
| Order p | 1 |
H(10,1) ≈ 2.928968254; last term = 0.1; average term ≈ 0.2928968254.
Assumptions and limits
- n is a positive whole number and p is positive; the result is a finite sum rather than an assertion about convergence to infinity.
- Each term uses the same dimensionless index k and the calculation uses ordinary real-number arithmetic.
- For large n or extreme p, displayed precision is limited by JavaScript number arithmetic even though the finite definition is explicit.
- The page does not replace a proof of convergence, asymptotic estimate, or infinite-series remainder bound.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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