[AntonioLao]

In the study of infinite series which are essential for studying the calculus, a long standing problem has been the summation of integer powers of the reciprocals of the positive integers known in general as harmonic series. For the special case where the integer power is unity the series is given by 1+½+1/3+¼+1/5+1/6+1/7+1/8+… More than 2300 years ago Aristotle believed it has a finite sum without knowing how to present a proof.

Eventually, this was proved in the year 1350, to be unbounded and without a limit, its sum diverges to infinity, by the medieval French mathematician and philosopher Nicole Oresme (1320-82). His simple proof was to group the series into its partial sums starting with (1/3+1/4), (1/5+1/6+1/7+1/, etc. and then replacing every term of every group by the smallest term in each group: (1/4+1/4) and (1/8+1/8+1/8+1/, etc. and by inspection it is easily noted that each of these partial sums adds up to ½ and 2 partial sums equal unity and infinite unities become an harmonic infinity. Consequently, the original bigger series must also sum to infinity. However, if the integer power is greater than unity, say 2 then the series is given by 1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+…or equivalently 1+1/4+1/9+1/16+1/25+1/36+1/49+1/64+…and in 1743 it was proved by Euler (1707-83) to converge to the finite sum p/6. In general, all harmonic series with integer power greater than unity converge to a finite sum. For power 4, the sum is p/90. For power 6, the sum is pp/945.

[Profpat]

Well so much for the harmony Antonio. If it's infinite no more harmony.

[mkirkpatrick]

Great thread starter Antonio,the whole universe dances to the tune of harmonics,without
it we could not exist.




regards michael.

[AntonioLao]

Originally Posted by mkirkpatrick

without it we could not exist.

Euler as the greatest analytical mathematician of all time failed to prove the existence of odd harmonies. His world-class tour de force was proving the even harmonies having a finite limit.

[mkirkpatrick]

Originally Posted by AntonioLao

Euler as the greatest analytical mathematician of all time failed to prove the existence of odd harmonies. His world-class tour de force was proving the even harmonies having a finite limit.

Thats true,however it is directly related to motion,motion=harmony=physical
universe.No motion no harmony no physical universe!Just the unmanifested
potential.



regards michael.

[AntonioLao]

Originally Posted by mkirkpatrick

it is directly related to motion

Actually, the theory that I am developing is that of a local infinitesimal motion relating the absolute acceleration to the local infinitesimal metrics and for mass unity this is the primary repulsive superforce of the quantum space-time.

[mkirkpatrick]

Originally Posted by AntonioLao

Actually, the theory that I am developing is that of a local infinitesimal motion relating the absolute acceleration to the local infinitesimal metrics and for mass unity this is the primary repulsive superforce of the quantum space-time.

Sounds very interesting,would very much like to see the end results of this development.



regards michael.

[AntonioLao]

Originally Posted by mkirkpatrick

the end results

It's trapped in an infinite loop without any exit.