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Maclaurin-Cauchy Theorem

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

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

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

\begin{displaymath}
\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),
\end{displaymath}

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




© 1996-9 Eric W. Weisstein
1999-05-26