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Topic: log n and sigma 1/n (Read 3236 times) |
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kaushiks.nitt
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log n and sigma 1/n
« on: Jun 13th, 2009, 10:28am » |
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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 .
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Obob
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Re: log n and sigma 1/n
« Reply #1 on: Jun 13th, 2009, 12:31pm » |
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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.
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Eigenray
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Re: log n and sigma 1/n
« Reply #2 on: Jun 13th, 2009, 1:31pm » |
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If you want to be picky we should use
1+1/2+...+1/n ~  + log(n+1/2)
for an O(1/n2) error, or more generally,
1+1/2+...+1/n ~ + log[ n+ 1/2 + 1/(24n) - 1/(48n3) + ... ]
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