Using the fractions.......................and combining numerator and denominator..............as I did for 1..........1/1 = 11..............................1/9 + 1/10.............a quantum leap happens...............1/9 = 19..................1/10 = 110.......................19 to 110..........a jump of 91...............like 19 backwards..........mirror symmetry..............reflexive symmetry..........................like wow........

Harmonic Series The series (1) is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function . The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire 2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter (Derbyshire 2004, pp. 9-10). Progressions of the form (2) are also sometimes called harmonic series (Beyer 1987). Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms (after the first two) and noting that each such block has a sum larger than 1/2, (3) (4) and since an infinite sum of 1/2's diverges, so does the harmonic series. The generalization of the harmonic series