Sunday, September 17, 2017

As for the rest of the harmonic series.............................many combinations of numerator and denominator become 11.................. as they do with double digit denominators, the 1st time after the fundamental ............using the zero dimension..........for single digit denominators it is different................1/2 + 1/3 + 1/4, etc..........becomes...........12 + 13 + 14, etc...............the first fraction, after the fundamental..........is 1/2........it becomes 12.........twelve........what 4 and 3 produce............the quantum leap from the other pairs......the 1st zero in the R zeta function is at double digit imaginary numbers..............14..i..............why might that be?  It is after the fundamental...................2 - 11......................13 is the closest prime to 14i........close does count Albert E....................................1 being the fundamental of the harmonic series..............can be seen as 11.......1 is 1/1, as well as many higher up as in 1/10, the next such instance..................which becomes.........110........jumps.........................they all don't...............now and then.........quantum leaps and chaos......combined

B/c with 1/20 it does not start with an 11.................1/20 becomes 120...........


Harmonic Series

DOWNLOAD Mathematica Notebook EXPLORE THIS TOPIC IN the MathWorld Classroom Contribute to this entry The series
 sum_(k=1)^infty1/k
(1)
is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function 1/x. 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
 1/(a_1),1/(a_1+d),1/(a_1+2d),...
(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,
sum_(k=1)^(infty)1/k=1+1/2+(1/3+1/4)+(1/5+1/6+1/7+1/8)+...
(3)
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