For $ n $ be a positive integer and $ f(n) = \dfrac{\pi^2(n)}{n} $, where $ \pi(x) $ is the number of the prime numbers less than $x$ . Then find the value of $\displaystyle S = \lim_{n \to \infty} \dfrac{1}{n} \left(\sum_{k=1}^{n} f(2k)\right).$
By the computation, it seems to be $ S=1 $. But I cannot prove this.