# 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

Thus, the $C_{n}$ decrease monotonically, while the $D_{n}$ increase monotonically, since the differences are negative (positive for $D_{n}$). Further, $D_{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

Title proof that Euler’s constant exists ProofThatEulersConstantExists 2013-03-22 16:34:48 2013-03-22 16:34:48 rm50 (10146) rm50 (10146) 6 rm50 (10146) Proof msc 40A25