2, 3, 5, 7, 11......1.......is also 1/1..........................which could be 11...........1/2.....could be 12..........1/3.....could be 13...............1/9.......19 and so forth.....part of the problem is that u people do not believe these things are connected this way.........and again.....do NOT believe me to believe me......CHECK.........................so a very important hidden thing is there............b/c there are jumps................the next fraction in the harmonic series after 1/9 is 1/10..........if u use 1/9 as 19............then 1/10 becomes 110...........a jump.....from 19 to 110............................the prime number theorem......Gauss noted a distribution in primes..............of 2.................but an erratic one.........it was a good ball park figure............but inside that distribution there was an erratic number..............i bet the harmonic series holds where the jumps are................
A big hint......the harmonic series itself........and an instrument......has a fraction......an overtone.......AND the fundamental....................which is always in the numerator.............
Comparison test[edit]
One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest
power of two:
![{\displaystyle {\begin{aligned}&1\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{3}}\,+\,{\frac {1}{4}}\;\;+\;\;{\frac {1}{5}}\,+\,{\frac {1}{6}}\,+\,{\frac {1}{7}}\,+\,{\frac {1}{8}}\;\;+\;\;{\frac {1}{9}}\,+\,\cdots \\[12pt]>\;\;\;&1\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{4}}\,+\,{\frac {1}{4}}\;\;+\;\;{\frac {1}{8}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{8}}\;\;+\;\;{\frac {1}{16}}\,+\,\cdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d33be1134374a704aed1d911722c4095554015c1)
Each term of the harmonic series is greater than or equal to the corresponding term of the second series, and therefore the sum of the harmonic series must be greater than the sum of the second series. However, the sum of the second series is infinite:
![{\displaystyle {\begin{aligned}&1+\left({\frac {1}{2}}\right)+\left({\frac {1}{4}}+{\frac {1}{4}}\right)+\left({\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}\right)+\left({\frac {1}{16}}+\cdots +{\frac {1}{16}}\right)+\cdots \\[12pt]=\;\;&1\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{2}}\;\;+\;\;{\frac {1}{2}}\;\;+\;\;\cdots \;\;=\;\;\infty \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59ba63c85f07a3cfbb009696fc135d757be3052)
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