Calculator Inputs
Standard harmonic numbers use m = 1, p = 1, a = 1, b = 0, with alternating unchecked.
Formula Used
General finite form:
S = a × Σ[σ(k) / (k + b)p], for k = m to n
Where:
- a = scaling coefficient
- m = starting index
- n = ending index
- b = shift applied to the denominator
- p = order of the generalized harmonic sum
- σ(k) = 1 for standard sums, or (-1)k-1 for alternating sums
For the standard harmonic number, set a = 1, m = 1, b = 0, p = 1, and do not alternate signs. Then the calculator evaluates Hn = 1 + 1/2 + 1/3 + ... + 1/n.
When the standard setting is chosen, the tool also compares the exact value with the classic approximation: Hn ≈ ln(n) + γ + 1/(2n) − 1/(12n2) + 1/(120n4), where γ is Euler’s constant.
How to Use This Calculator
- Enter the starting term m and ending term n.
- Set the order p. Use 1 for the usual harmonic sum.
- Choose coefficient a to scale each term.
- Add a shift b only when you need a shifted denominator.
- Enable alternating signs if you want alternating harmonic behavior.
- Pick your preferred decimal places.
- Press the calculate button to place the result above the form.
- Review the summary cards, graph, and detailed term table.
- Use the CSV or PDF buttons to export your result.
Example Data Table
Sample standard harmonic values from H1 through H10.
| n | Added Term | Harmonic Number Hn |
|---|---|---|
| 1 | 1.000000 | 1.000000 |
| 2 | 0.500000 | 1.500000 |
| 3 | 0.333333 | 1.833333 |
| 4 | 0.250000 | 2.083333 |
| 5 | 0.200000 | 2.283333 |
| 6 | 0.166667 | 2.450000 |
| 7 | 0.142857 | 2.592857 |
| 8 | 0.125000 | 2.717857 |
| 9 | 0.111111 | 2.828968 |
| 10 | 0.100000 | 2.928968 |
Frequently Asked Questions
1. What does this calculator compute?
It computes finite harmonic and generalized harmonic sums over a chosen index range. It can also apply a coefficient, denominator shift, and alternating signs.
2. What is the standard harmonic series?
The standard harmonic series uses terms 1/k with k starting at 1. Its finite partial sums are harmonic numbers, while the full infinite series diverges.
3. What does order p change?
The order p changes the denominator power. When p equals 1, you get the usual harmonic pattern. Larger p values make terms shrink faster.
4. When should I use the alternating option?
Use it when successive terms should flip signs. This is useful for alternating harmonic-style sums and for comparing conditional versus absolute convergence behavior.
5. Why does the tool show an approximation only sometimes?
The approximation is shown only for the standard non-alternating harmonic setting. That is where the Euler-log expansion is directly appropriate and meaningful.
6. What does the shift value b do?
It changes each denominator from k to k + b. This helps model shifted reciprocal sums, but it must not create a zero denominator.
7. Why is the number of terms limited?
The limit keeps the page responsive for tables, exports, and graph drawing. It also helps prevent extremely large browser downloads and slow rendering.
8. What can I export from this page?
You can export the calculated term table as CSV and save the visible result section as a PDF. Both options appear after calculation.