(1)Is this a unsolved problem ?
(2)Then what is the good approximate answer to S(n) decimal places .?
(3) If some one have proof of approximate equation to S(n) ,then it is a ground braking work ?
(4) I Have a pure mathematical approximation ( correct to 4 decimal places) to compute it for all possible value of n !
I have been prove that S(n) for each integer have approximation correct 4 decimal places have close form as
S(n) = ln | (2n+1)/2 | + Euler constant
Euler constan = 0.57721566....
I HAVE THE POOF ...!
EX-
S(10)= 1+1/2+1/3+.......+1/10 = 2.928584438..............
AND
S(10) = ln | (20+1)/2 | + Euler constant = 2.928590922.........
Dear Manoj, your own discovery will remain yours, even if another person would discover the same (before). Keep your enthusiasm for science :)
Your result looks interesting, however it would be good to improve it. For small n, the approximation is not sufficiently precise. For large values, it gets fine (better and better with growing n). As George suggests, do not write too much publically (here on RG) before publishing the result - it is always possible to post the work in ArXiv, etc.
S(n) is called a harmonic number, usually denoted Hn. I encourage you to focus also Digamma function, Gamma function, ... For an integral representation of Hn and other interesting results, see https://arxiv.org/abs/math/0506319
This harmonic series does not approach a limit as n increases. A partial sum can be constructed from 1/n + 1/(n-1) + ...+ 1/2 +1 for any n. Each partial sum is a fraction with n terms in the nominator and one term in the denominator which is a multiple of the series denominators.
The solution depend on the decimal places and the convergence term you want...
Dear Manoj, inequality (15) in http://mathworld.wolfram.com/HarmonicNumber.html is close to your result.
It can be useful to read the works cited therein, where (15) is proved [DeTemple 1991; Havil 2003].
If your estimation is better, or your proof is different, it is still worth to continue. Good luck!
With your best encouragements of all kind of mathematicians .., I able to improve the approximation and compute value is correct 5 decimal places now
H10 = 2.92897...........
@ Viera Čerňanová
My method is different than ,
http://mathworld.wolfram.com/HarmonicNumber.html . and i would like to gives you special thank to finds me the best information about Harmonic series.
thank you
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