Can $\{n!\log n\}$ converge to $1$?

Question

Since $H_n-\log n\to\gamma$ is it correct to deduce that, if $\gamma$ were a rational $a/b $, then $$\lim_{n\to \infty }\{n!H_n-n!\log n\}=\lim_{n\to \infty }\{-n!\log n\}=1-\lim_{n\to \infty }\{n!\log n\}=0$$? If the implication is correct, what is known about $\{n!\log n\}$?

Answer 1

There appear to be two serious errors of thought here.

The first is that $\{ \cdot \} $ is not a continuous function. So, the fact that $a_n \to L$ does not mean you can clude $\{ a_n \} \to \{ L \}$. This would be true if $L$ wasn't an integer, but you are specifically interested about the case where it is.

The second problem is that you make absolutely no attempt to control the error of the approximation. In asymptotic notation, that $H_n - \log n \to \gamma$ means

$$ H_n - \log n = \gamma + o(1) $$

multiplying by $n$ gives

$$ n! H_n - n! \log n = \gamma n! + o(n!) $$

The problem is the error term $o(n!)$ is much too large for you to say anything about the fractional part of the right hand side.