## Goldie Horton (University of Texas)

### 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 $\log\left(\left(\frac{n+1}{n+1}\right)^{n+2}\left(\frac{n+1}{n}\right)^{n+1}\right).$