Upper bound on $\sum_{n=1}^N \frac{1}{n}$ are there any analytical upper bounds on
\begin{align}
\sum_{n=1}^N \frac{1}{n}
\end{align}
Clearly, one upper bound is $N$. But there has to be a better one. There is an approach with Harmonic number but it's not "analytic".
Thank you

$\sum_{n=1}^N {1 \over n} \le 1+ \ln N$. 
This follows from $\int_{n-1}^n {1 \over t} dt = \ln n - \ln (n-1) > {1 \over n}$.
