proof that Euler’s constant exists

Theorem 1

The limit

\gamma=\lim_{n\to\infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\ln n\right)

exists.

Proof. Let

C_{n}=\frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n

and

D_{n}=C_{n}-\frac{1}{n}

Then

C_{n+1}-C_{n}=\frac{1}{n+1}-\ln\left(1+\frac{1}{n}\right)

and

D_{n+1}-D_{n}=\frac{1}{n}-\ln\left(1+\frac{1}{n}\right)

Now, by considering the Taylor series for $\ln(1+x)$ , we see that

\frac{1}{n+1}<\ln\left(1+\frac{1}{n}\right)<\frac{1}{n}

and so

C_{n+1}-C_{n}<0<D_{n+1}-D_{n}

Thus, the $C_{n}$ decrease monotonically, while the $D_{n}$ increase monotonically, since the differences are negative (positive for $D_{n}$ ). Further, $D_{n}<C_{n}$ and thus $D_{1}=0$ is a lower bound for $C_{n}$ . Thus the $C_{n}$ are monotonically decreasing and bounded below, so they must converge.

References

1 E. Artin, The Gamma Function, Holt, Rinehart, Winston 1964.

Title	proof that Euler’s constant exists
Canonical name	ProofThatEulersConstantExists
Date of creation	2013-03-22 16:34:48
Last modified on	2013-03-22 16:34:48
Owner	rm50 (10146)
Last modified by	rm50 (10146)
Numerical id	6
Author	rm50 (10146)
Entry type	Proof
Classification	msc 40A25