Is the sequence $\log(n) - \sum_{k=1}^{n-1} \frac{1}{k}$ bounded?

Question

Been looking at the sequence $\big{(}\log(n) - \sum_{k=1}^{n-1} \frac{1}{k}\big{)}$, it seems like its bounded but I am not sure if it is. Is it? and if so is this a result I can see a reference for?

Answer 1

Hint: Notice that $$ log(n)=\int_{1}^{n}\frac{1}{x} $$ Now look at the sum and think about how the Riemann-Darboux integral.

Let me know if you need further help

Answer 2

Using harmonic numbers $$\Delta_n=\log(n) - \sum_{k=1}^{n-1} \frac{1}{k}=\log(n)-H_{n-1}$$ For large values of $p$ $$H_p=\log (p)+\gamma +\frac{1}{2 p}+O\left(\frac{1}{p^2}\right)$$ $$\Delta_n=-\gamma +\log \left(\frac{n}{n-1}\right)-\frac{1}{2 (n-1)}+\cdots$$