- #1
Yuqing
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The definition of the Euler - Mascheroni constant, [itex]\gamma[/itex], is given as
[tex]\gamma = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k} - \ln(n)[/tex]
or equivalently in integral form as [tex]\gamma = \int_{1}^{\infty}\frac{1}{\left\lfloor x\right\rfloor} - \frac{1}{x}\ dx[/tex]
I saw a seeming related integral representation
[tex]\gamma = 1 - \int_{1}^{\infty} \frac{x - \left\lfloor x\right\rfloor}{x^2}\ dx[/tex]
but I can't seem to derive it. I was wondering if anyone can shed some light on this.
[tex]\gamma = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k} - \ln(n)[/tex]
or equivalently in integral form as [tex]\gamma = \int_{1}^{\infty}\frac{1}{\left\lfloor x\right\rfloor} - \frac{1}{x}\ dx[/tex]
I saw a seeming related integral representation
[tex]\gamma = 1 - \int_{1}^{\infty} \frac{x - \left\lfloor x\right\rfloor}{x^2}\ dx[/tex]
but I can't seem to derive it. I was wondering if anyone can shed some light on this.