Proof of Euler-Mascheroni Constant

[coki2000]

Hi PF,

I have question about harmonic series.

\lim_{n\to \infty}(\sum_{k=1}^{\infty}\frac{1}{k}-ln(n))=ln(n)+\gamma +\epsilon _0

I couldn't find any proof for that equality. Do you know its exact proof?

Thanks for helps.

[dextercioby]

The formula is wrong. It should read sum to n in the brackets and no ln(n) in the RHS.

[coki2000]

Oh, sorry you're right it should be like that

\sum_{k=1}^{n}\frac{1}{k}=ln(n)+\gamma +\epsilon _n

My question is where it comes from? How can we prove it?
Thanks for your correction.

[Citan Uzuki]

Essentially, it comes from the fact that γ is defined to be \lim_{n \rightarrow \infty}\sum_{k=1}^n \frac{1}{k} - \ln(n), so the only thing that is necessary is to prove that this limit actually exists and is finite. For this, consider the following:

\begin{align*} \sum_{k=1}^{n} \frac{1}{k} - \ln(n) &= \sum_{k=1}^{n} \int_{k}^{k+1}\frac{1}{\lfloor x \rfloor} \ dx - \int_{1}^{n} \frac{1}{x} \\ &= \int_{1}^{n}\frac{1}{\lfloor x \rfloor} - \frac{1}{x} \ dx + \int_{n}^{n+1} \frac{1}{\lfloor x \rfloor} \ dx \\ &= \int_{1}^{n} \frac{x-\lfloor x \rfloor}{x \lfloor x \rfloor} \ dx + \frac{1}{n} \end{align*}

Now, as n→∞, 1/n → 0, and the integral on the left converges to:

\int_{1}^{\infty} \frac{x-\lfloor x \rfloor}{x \lfloor x \rfloor} \ dx

But this integral is finite, since:

\frac{x-\lfloor x \rfloor}{x \lfloor x \rfloor} \leq \frac{1}{\lfloor x \rfloor^{2}}

And:

\int_{1}^{\infty} \frac{1}{\lfloor x \rfloor^{2}} \ dx = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} < \infty

[coki2000]

@Citan Uzuki

Thank you very much for your enlightening response. But what about the term \epsilon_n. Where it comes from? And also how can we calculate γ numerically?

[dextercioby]

The epsilon is a very small number accounting for the error which disappears when taking the limit.

[coki2000]

The epsilon is a very small number accounting for the error which disappears when taking the limit.
In this article (http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Rate_of_divergence), it says that \epsilon_k is approximately 1/2k so I wonder why Euler inserted this factor into this equality and how he found that it is approximately 1/2k(not 1/5k for example). And is there any special name for that epsilon factor?

Thanks for all your helps :)