# Upper bound for $n \sum_{k=1}^n \frac1k$

What is upper bound of sum $$n \sum_{k=1}^n \frac1k$$ . Is it $$O(n\log n)$$ (i.e sum is bounded by $$c\cdot n\log n$$ for some constant $$c$$ and for all $$n \ge n_0$$) ? How to prove it ?

We have $$n\sum_{k=1}^n\frac 1k\le n\cdot\left(1+ \sum_{k=2}^n\int_{k-1}^k\frac {\mathrm dx}x\right)=n\left(1+\int_1^n\frac {\mathrm dx}x\right)=n(1+\ln n)<2n\ln n$$ for all $$n>e$$.
Look at Harmonic numbers and note that the partial sums $$H_n = \sum_{k=1}^n \frac1k$$ can be approximated by $$\ln n + \gamma$$, where $$\gamma$$ is the Mascheroni constant.