Harmonic series (mathematics)
| n | Partial sum of the harmonic series, Hn | |||
|---|---|---|---|---|
| expressed as a fraction | decimal | relative size | ||
| 1 | 1 | 1 |
|
|
| 2 | 3 | /2 | 1.5 |
|
| 3 | 11 | /6 | ~1.83333 |
|
| 4 | 25 | /12 | ~2.08333 |
|
| 5 | 137 | /60 | ~2.28333 |
|
| 6 | 49 | /20 | 2.45 |
|
| 7 | 363 | /140 | ~2.59286 |
|
| 8 | 761 | /280 | ~2.71786 |
|
| 9 | 7129 | /2520 | ~2.82897 |
|
| 10 | 7381 | /2520 | ~2.92897 |
|
| 11 | 83711 | /27720 | ~3.01988 |
|
| 12 | 86021 | /27720 | ~3.10321 |
|
| 13 | 1145993 | /360360 | ~3.18013 |
|
| 14 | 1171733 | /360360 | ~3.25156 |
|
| 15 | 1195757 | /360360 | ~3.31823 |
|
| 16 | 2436559 | /720720 | ~3.38073 |
|
| 17 | 42142223 | /12252240 | ~3.43955 |
|
| 18 | 14274301 | /4084080 | ~3.49511 |
|
| 19 | 275295799 | /77597520 | ~3.54774 |
|
| 20 | 55835135 | /15519504 | ~3.59774 |
|
| 21 | 18858053 | /5173168 | ~3.64536 |
|
| 22 | 19093197 | /5173168 | ~3.69081 |
|
| 23 | 444316699 | /118982864 | ~3.73429 |
|
| 24 | 1347822955 | /356948592 | ~3.77596 |
|
| 25 | 34052522467 | /8923714800 | ~3.81596 |
|
| 26 | 34395742267 | /8923714800 | ~3.85442 |
|
| 27 | 312536252003 | /80313433200 | ~3.89146 |
|
| 28 | 315404588903 | /80313433200 | ~3.92717 |
|
| 29 | 9227046511387 | /2329089562800 | ~3.96165 |
|
| 30 | 9304682830147 | /2329089562800 | ~3.99499 |
|
In mathematics, the harmonic series is the divergent infinite series:
Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music.
The fact that the harmonic series diverges was first proven in the 14th century by Nicole Oresme, but this achievement fell into obscurity. Proofs were given in the 17th century by Pietro Mengoli,Johann Bernoulli, and Jacob Bernoulli.
Historically, harmonic sequences have had a certain popularity with architects. This was so particularly in the Baroque period, when architects used them to establish the proportions of floor plans, of elevations, and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.
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Wikipedia