Why is the sum of the factorials of each of the values in the harmonic series almost equivalent to the upper bound of the summation?

Question

I was wondering why the functions, $f(x) = \sum_{i=1}^x \frac{1}{i}!$ and $ g(x) = x $ are almost equivalent when graphed. Is this just a coincidence, or is there more to it?

Answer 1

$x! = \Gamma(1 + x)$ has a Taylor series expansion $1 - \gamma x + O(x^2)$ which gives

$$\sum_{i=1}^n \Gamma \left( 1 + \frac{1}{n} \right) = n - \gamma \log n + O(1).$$

So it is asymptotic to $n$ but there's an additive error of about $\gamma \log n$.