Euler's constant

Euler's constant $\gamma$ is defined by

$$ \gamma = \lim_{n\rightarrow \infty}\; \left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln {n}\right) $$

or equivalently

$$ \gamma = \lim_{n\rightarrow \infty}\; \sum_{i=1}^n \left[\frac{1}{i} - \ln \left( 1 + \frac{1}{i} \right) \right] $$

Euler's constant has the value

$$ 0.57721566490153286060651209008240243104\ldots $$

It is related to the gamma function by

$$ \gamma = - \Gamma'(1) $$

It is not known whether $\gamma$ is rational or irrational.

References.

• Chris Caldwell - ``Euler's Constant'', http://primes.utm.edu/glossary/page.php/Gamma.html