Existence of Euler-Mascheroni Constant

Theorem

The sequence:

$\displaystyle \left \langle \sum_{k=1}^n \frac 1 k - \ln n \right \rangle$

converges to a limit.

This limit is known as the Euler-Mascheroni constant.

Proof

Let $f:\R \to \R, \displaystyle f \left({x}\right) = \frac 1 x$.

Clearly $f$ is continuous, positive and decreasing on $\left[{1 \,.\,.\, \infty}\right)$.

Therefore the conditions of the Euler-Maclaurin Summation Formula hold.

Thus the sequence $\left \langle {\Delta_n} \right \rangle$ defined as:

$\displaystyle \Delta_n = \sum_{k=1}^n f \left({k}\right) - \int_1^n f \left({x}\right) \ \mathrm d x$

is decreasing and bounded below by zero.

But from the definition of the natural logarithm:

$\displaystyle \int_1^n \frac {\mathrm d x} x = \ln n$

Hence the result.

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 14.3 \ (5)$