In some of my number theoretical calculations I saw the sequence
$$a_ N = \frac{\sum_{j=1}^{N} \{\frac{N}{j}\} }{N}$$
for $1\leq N$ and $\{x\}$ means the fractional part of $x$.
I checked with a computer and I found out that the sequence converges to a number near $0.422774$. How to prove the convergence and what is this constant? It should be well-known and should be some relations between it and some other number theoretical constants like $\gamma$ (Euler-Mascheroni Constant).
With the advise submitted by Thomas Andrews the written form isn't any thing except a Riemann sum for calculating the integral
$$\int_{0}^1 \left\{\frac{1}{x}\right\}\,dx.$$
Thus we should calulate this integral. we have:
$$\int_{0}^1 \left\{\frac{1}{x}\right\}\,dx = \sum _{n=1}^\infty \int_{\frac{1}{n+1} }^{\frac{1}{n} } \left\{\frac{1}{x}\right\}\,dx = \sum _{n=1}^\infty \int_{\frac{1}{n+1} }^{\frac{1}{n} } \Big(\frac{1}{x} - n \Big) \,dx = \sum _{n=1}^\infty \Big(\log(n+1) - \log(n) - \frac{1}{n+1} \Big ) = 1 - \gamma $$
which the $\gamma $ is the Euler-Mascheroni constant.