Euler-Mascheroni Constant

So n is a natural number, so x should be an integer.

Aaah! No. The notation $\displaystyle\sum_0^{\infty}$ is a limit: $\displaystyle\sum_0^{\infty}= \lim_{\infty}\sum_0^n$

(the same can be said about improper integrals - writing the infinity symbol directly is a shorthand; mathematicians are lazy!)
For the limit to make sense, the initial expression must make sense: $\displaystyle\sum_0^n$ only makes sense if $n\in\mathbb{N}$

I know the notation.

I am so confused right now.

Summation implies n must be natural number, tho' I can probably define a summation close to an integral with infinitesimal interval.

If n is a natural number, the entire thing will be integer.

I don't know how to make it clearer. $\displaystyle\sum_0^{n} k$ is an integer, $\displaystyle\lim_{n\to\infty} \sum_0^{n} k$ is not.

Yes $n$ is an integer, but we are letting it tend to infinity, the expression diverges, and thus the limit is not an integer.

I see what you mean, but you probably misunderstood.

I mean, if n is bounded to be a natural number, won't n still be a natural number even n tends to infinity?

EDIT: And the term is $n! \sum_{1\le k\le n} \frac{1}{k}$ and after summing all fractions together, the denominator will be $n!$, so it will be cancelled out.

Define $x=\displaystyle\lim_{n\to \infty} \underbrace{b (\gamma-\sum_{1\le k \le b}\frac{1}{k}+\ln b) b!}_{\text{fixed real number}} \cdot\underbrace{n! (n-1)^{\frac{n(n+1)}{2}}}_{\text{ divergent sequence!}}$

Also, I find it a bit surprising that you aren't questioning yourself given that this number appeared in 1743, that the greatest mathematicians in history have lived since then, and that it still has not been proven to be irrational. Not saying it will never be done, but don't you think the person who concocted that particular proof of the irrationality of e would have been clever enough to think "hm, I could use the same method to prove the irrationality of the Euler-Mascheroni constant too!"

Not saying it will never be done, but don't you think the person who concocted that particular proof of the irrationality of e would have been clever enough to think "hm, I could use the same method to prove the irrationality of the Euler-Mascheroni constant too!"

Notably, the first person to prove $e$ irrational being Euler himself.

Yep, I was referring to Fourier's. Don't think I've ever seen Euler's.