The harmonic series is a well-known divergent infinite series that is generally used in many areas of mathematics and physics. Though it look simple, but reveals deep mathematical concepts for scientists.
The harmonic series is defined as the infinite sum:
Usually, the partial sum up to n terms is denoted as:
Harmonic Series Calculator
Calculate the partial sum of the harmonic series up to any positive integer n:
Enter a positive integer to calculate the sum of the harmonic series up to that number.
Mathematical Properties
Divergence
The harmonic series is different because of its divergent nature i.e. its sum nearly approaches infinity as more terms are added. This condition is proven by Nicole Oresme in the 14th century using a clever grouping argument.
1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + …
1 + 1/2 + (2/4) + (4/8) + …
1 + 1/2 + 1/2 + 1/2 + …
Each grouped term sums to at least 1/2, showing the series must diverge.
Rate of Growth
Although, the harmonic series diverges, it does so very slowly. The partial sum Hn grows approximately at the rate of ln(n) + γ, where γ ≈ 0.57721 is the Euler-Mascheroni constant.
Hn ≈ ln(n) + γ + 1/(2n) – 1/(12n²) + …
For example, it takes over 12,000 terms just to reach a sum of 10!
Interesting Facts of Harmonic Calculator
Harmonic Distance
If you place blocks at positions 1, 1/2, 1/3, 1/4, etc., the stack will eventually exceed any height, but you would need an enormous number of blocks.
Alternating Harmonic Series
Unlike the standard harmonic series, the alternating harmonic series (1 – 1/2 + 1/3 – 1/4 + …) converges to ln(2) ≈ 0.693.
Historical Note
The term “harmonic” comes from music theory, where strings of lengths proportional to 1, 1/2, 1/3, etc. produce harmonious tones.
Riemann Zeta Function
The harmonic series is a special case of the Riemann zeta function: ζ(1) = H∞, which is crucial in number theory and the study of prime numbers.
Related Series
| Series | Formula | Convergence | Sum (if convergent) |
|---|---|---|---|
| Harmonic Series | Σ 1/n | Diverges | ∞ |
| Alternating Harmonic | Σ (-1)n+1/n | Converges | ln(2) ≈ 0.693 |
| p-Series (p=2) | Σ 1/n² | Converges | π²/6 ≈ 1.645 |
| p-Series (general) | Σ 1/np | Converges if p > 1 | ζ(p) (Riemann zeta function) |
Did You Know?
Harmonic Numbers and Approximation: For large values of n, the harmonic number Hn can be approximated by:
Where γ ≈ 0.57721 is the Euler-Mascheroni constant. This approximation becomes more accurate as n increases.
Harmonic Series Milestones
| n | Approximate Value of Hn | Milestone |
|---|---|---|
| 10 | 2.9290 | First few terms |
| 100 | 5.1874 | Hundred terms |
| 1,000 | 7.4855 | Thousand terms |
| 10,000 | 9.7876 | Ten thousand terms |
| 10100 | ≈ 230.8 | Googol terms! |