proof that Euler’s constant exists
Theorem 1
The limit
$$\gamma =\underset{n\to \mathrm{\infty}}{lim}\left(\sum _{k=1}^{n}\frac{1}{k}-\mathrm{ln}n\right)$$ |
exists.
Proof. Let
$${C}_{n}=\frac{1}{1}+\frac{1}{2}+\mathrm{\cdots}+\frac{1}{n}-\mathrm{ln}n$$ |
and
$${D}_{n}={C}_{n}-\frac{1}{n}$$ |
Then
$${C}_{n+1}-{C}_{n}=\frac{1}{n+1}-\mathrm{ln}\left(1+\frac{1}{n}\right)$$ |
and
$${D}_{n+1}-{D}_{n}=\frac{1}{n}-\mathrm{ln}\left(1+\frac{1}{n}\right)$$ |
Now, by considering the Taylor series^{} for $\mathrm{ln}(1+x)$, we see that
$$ |
and so
$$ |
Thus, the ${C}_{n}$ decrease monotonically, while the ${D}_{n}$ increase monotonically, since the differences are negative (positive for ${D}_{n}$). Further, $$ 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 |