  For $n\ge 1$, the $n^{\rm th}$ Ramanujan prime is defined as the smallest
positive integer $R_n$ such that for all $x\ge R_n$, the interval
$(\frac{x}{2}, x]$ has at least $n$ primes. We show that for every
$\epsilon>0$, there is a positive integer $N$ such that if
$\alpha=2n\left(1+\dfrac{\log 2+\epsilon}{\log n+j(n)}\right)$, then $R_n<
p_{[\alpha]}$ for all $n>N$, where $p_i$ is the $i^{\rm th}$ prime and $j(n)>0$
is any function that satisfies $j(n)\to \infty$ and $nj'(n)\to 0$.
