Math Help - Euler–Mascheroni constant

1. Euler–Mascheroni constant

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 .

2. Originally Posted by kaushiks.nitt
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 .

Euler-Mascheroni Constant -- from Wolfram MathWorld

Euler?Mascheroni constant - Wikipedia, the free encyclopedia

Applications of the Euler–Mascheroni constant to physics are numerous eg. Bessel's and Helmholtz' equations (heat conduction, electromagnetic waves, etc.), regularization of Feynman diagrams in QFT etc. etc.