Consider the sequence $\lbrace \frac{\phi(i)}{i}\rbrace\_{i=1}^\infty$ where $\phi$ is the Euler's function. The Sequence is clearly dense in $[0,1]$. What can be said about the limsup of its average sequence ?? I mean the sequence $\frac 1n\sum\_{i=1}^n \frac{\phi(i)}{i}$. Its value or an upper bound would be helpful. 



A Sequence of Real numbers 
It is well known that $\sum\_{k\le x} \phi(k)=\frac{3}{\pi^2}x^2+O(x\log(x))$. In a similar way we obtain
$$
\sum\_{k\le x} \frac{\phi(k)}{k}=\frac{6}{\pi^2}x+O(\log(x)),
$$
by using the Moebius function, i.e., $\sum\_{k\le x} \frac{\phi(k)}{k}=\sum\_{k\le x}\sum\_{d\mid k}\frac{\mu(d)}{d}=\sum\_{d\le x}\frac{\mu(d)}{d}\lfloor \frac{x}{d}\rfloor$. Then
$$
\sum\_{k\le x} \frac{\phi(k)}{k}=x\sum\_{d\le x}\frac{\mu(d)}{d^2}-\sum\_{d\le x}\frac{\mu(d)}{d}((x/d))=\frac{6}{\pi^2}x+O(\log(x)).
$$


Edit: The term $O(1)$ (which I hade before from the paper of Carella) is not correct, please see the valuable comment of Peter.   

The value $6/\pi^2$ is reasonably good for the question posed, if $n$ is large.


