Euler’s constant

Euler’s constant $\gamma$ is defined by

$$\gamma =\underset{n\to \mathrm{\infty }}{lim}\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{\cdots }+\frac{1}{n}-\ln n\right)$$

or equivalently

$$\gamma =\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^n\left[\frac{1}{i}-\ln \left(1+\frac{1}{i}\right)\right]$$

Euler’s constant has the value

$$0.57721566490153286060651209008240243104\mathrm{\dots }$$

It is related to the gamma function by

$$\gamma =-{\mathrm{\Gamma }}^{\prime }(1)$$

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

References.

• Chris Caldwell - “Euler’s Constant”, http://primes.utm.edu/glossary/page.php/Gamma.htmlhttp://primes.utm.edu/glossary/page.php/Gamma.html