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        riddles >> hard >> log n and sigma 1/n
        
(Message started by: kaushiks.nitt on Jun 13^th, 2009, 10:28am)

Title: log n and sigma 1/n
Post by kaushiks.nitt on Jun 13^th, 2009, 10:28am

I have this doubt regarding the Euler–Mascheroni constant γ . It is integral 1/[x] - 1/x limits from 1 to infinity and it is value is
0.57721 . Of cousre the big question that remains is if this is rational or not .
Let us say i keep splitting the limits i.e from 1 to 2 then 2 to 3 and so on from n to n+1 . I use limit n -> infinity.
The value would be like this 1 - (log 2 - log1 ) + 1/2 - ( log 3 - log 2) + ... 1/n - ( log (n+1) - log n ) = γ.
Firstly sigma 1/n should have been approx to log ( n+1) rather than log(n) .
In such a case the error would be .5 .
So why do we approx to me log(n) and what is the specialty of the constant .

Title: Re: log n and sigma 1/n
Post by Obob on Jun 13^th, 2009, 12:31pm

The limit of (1 + 1/2 + ... + 1/n) - log n as n -> infty is the same thing as the limit of (1 + 1/2 + ... + 1/n) - log(n+1) as n -> infty, since the limit of log n - log(n+1) is 0 as n -> infty. So it doesn't make any difference either way.

Title: Re: log n and sigma 1/n
Post by Eigenray on Jun 13^th, 2009, 1:31pm

If you want to be picky we should use
1+1/2+...+1/n ~ http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/gamma.gif + log(n+1/2)
for an O(1/n²) error, or [link=http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant#Asymptotic_expansions]more generally[/link],
1+1/2+...+1/n ~ http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/gamma.gif + log[ n+ 1/2 + 1/(24n) - 1/(48n³) + ... ]