It matters a lot how you look at stuff.........14i........like 1/4th??  1/4 = 0.25............two seen primes............2 unseen primes..........like imaginary primes......shadow primes.........but with addition and subtraction........the other two come out...............5 - 2 = 3...............5 + 2 = 7...............2 and 5........seen...........3 and 7.......unseen..........all four................are exactly all four of single digit primes.........2, 3, 5, 7............there are only 4 single digit primes period...............1, 3, 7, 9...............are the end digits to how any prime >  10 can end.......................so a prime.........has a 25% chance of landing on a 1...........a 25% chance of landing on 3...................etc........above 10............25%..like 1/4th........................it matches up so well...........if it is a coincidence.........it is a great coincidence............

1/4th...........is the 4th term of the harmonic series..............but the 3rd fraction..............3rd fraction.....is 1/4.............a garbled pi.........I bet the entire fractal code is within the 1st 25 primes..................if u combined that with e and pi............and the next prime........over 100..........which surprisingly enough.....is 4 from the last double digit prime.......97...........which matches the four end digits.....possible end digits..........half of them..........1, 3, 7, 9..............reflex sym........with 7 and 9...............16 either way.............like the gold ratio.................7 + 9 = 16............9 + 7 = 16.........16 + 16 = 32..................an inverted......2, 3..........the start of all primes............

But that does match..............exactly with the 5 primes centered about 50..........from 40 - 60 are 5 primes..................41, 43, 47, 53, 59...........5 centered about 50..........5 * 50 = 250..........fractal patterns.........galore 007..........................if u break it down into < 50 and > 50...........(41, 43, 47)..............are all below 50............(53, 59).............above 50...............3 below 50, 2 above 50.............3, 2...................a pattern................also........using 50 as a guide........b/c it is 1/2 way from 0 - 100............or 1  -100...................either way.........0 and 1...........1/2 being dead center.......maybe u don't see it..........fractal patterns.................................

Divergence[edit]

There are several well-known proofs of the divergence of the harmonic series. A few of them are given below.

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:

${\displaystyle {\begin{aligned}&{}1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}+\cdots \\[12pt]>{}&1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots \end{aligned}}}$