Maclaurin-Cauchy Theorem

If $f(x)$ is Positive and decreases to 0, then an Euler Constant

$$\gamma_f \equiv \lim_{n\to\infty} \left[{\sum_{k=1}^n f(k)-\int_a^n f(x)\,dx}\right]$$

can be defined. If $f(x)=1/x$, then

$$\gamma = \lim_{n\to\infty} \left({\,\sum_{k=1}^n {1\over k}-... ...m_{n\to\infty} \left({\,\sum_{k=1}^n {1\over k}-\ln n}\right),$$

where $\gamma$ is the Euler-Mascheroni Constant.