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

I think the standard answer is $$\ln N+\gamma+\frac{1}{2N}$$ But I believe that with some effort that can be improved to $$\ln N+\gamma+\frac{1}{2N+\frac{1}{3}}$$
