Euler’s constant

Definition

Euler's constant, denoted by $\gamma$, is the limiting difference between the harmonic series and the natural logarithm. It approximately equals 0.57721.

5 Must Know Facts For Your Next Test

1. Euler's constant is defined as $\gamma = \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{1}{k} - \ln n \right)$.
2. It is not known whether $\gamma$ is rational or irrational.
3. Euler's constant appears in various areas of mathematics, including number theory and analysis.
4. $\gamma$ can be expressed through an integral: $\gamma = -\int_{0}^{\infty} e^{-x} \ln x \, dx$.
5. There are several series representations for $\gamma$, such as $\gamma = \sum_{k=1}^{\infty} \left( \frac{1}{k} - \ln(1 + \frac{1}{k}) \right)$.

Review Questions

Related terms

Harmonic Series:

The harmonic series is the divergent infinite series given by $H_n = \sum_{k=1}^{n} \frac{1}{k}$.

Natural Logarithm:

The natural logarithm, denoted as $\ln(x)$, is the logarithm to the base $e$, where $e$ is approximately equal to 2.71828.

Divergent Series: A divergent series is an infinite series that does not converge to a finite limit.