The number on top is always one in the harmonic series.............and the number on the bottom becomes key........jumps................1/9 then there is 1/10.........19 to 110...........double digit......to triple digit.............................1/99.......the bottom number is still.....double digit........the next fraction in the h. series.....is 1/100..............so if 199.....is from 1/99............then 1,100 is from 1/100..........a jump......from 199 to 1,100................and i think the key to understanding better the distribution of primes........which follow a basic pattern..........but is erratic inside it...........................
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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