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A note on the calculation of Euler's constant

American Mathematical Monthly, Vol. 23 (1916), p73.

Euler's constant is usually defined by the relation
\[\gamma = \lim_{n \rightarrow \infty} \sum_{m=1}^n \left[ \frac{1}{m}-\log\left(1+\frac{1}{m}\right) \right] \]

In this note Horton calls attention to the fact that Euler's constant
can be calculated to several places of accuracy from an idea used in
Cauchy's integral test for a convergent series of positive monotone
decreasing terms, namely that both
\[\sum_{m=1}^n u_m + \int_n^\infty u_m dm \quad \text{and} \quad \sum_{m=1}^n u_m + \int_{n+1}^\infty u_m dm \]

approximate the sum of the series with an error less than \( \int_n^{n+1} u_m dm\).
Horton uses this to show
that
\[\sum_{m=1}^n \left[ \frac{1}{m}-\log\left(1+\frac{1}{m}\right) \right] + \left[-1-\log\left(\frac{n}{n+1}\right)^{n+1}\right] \]

approximates Euler's constant with an error less than
\[\log\left(\left(\frac{n+1}{n+1}\right)^{n+2}\left(\frac{n+1}{n}\right)^{n+1}\right).\]