Existence of Euler-Mascheroni Constant

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Theorem

The real sequence:

$\displaystyle \sequence {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}$

converges to a limit.


This limit is known as the Euler-Mascheroni constant.


Proof

Let $f: \R \setminus \set 0 \to \R: \map f x = \dfrac 1 x$.

Clearly $f$ is continuous and positive on $\hointr 1 {+\infty}$.

From Reciprocal Sequence is Strictly Decreasing, $f$ is decreasing on $\hointr 1 {+\infty}$.

Therefore the conditions of the Integral Test hold.

Thus the sequence $\sequence {\Delta_n}$ defined as:

$\displaystyle \Delta_n = \sum_{k \mathop = 1}^n \map f k - \int_1^n \map f x \rd x$

is decreasing and bounded below by zero.

But from the definition of the natural logarithm:

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

Hence the result.

$\blacksquare$


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