Putnam 1976 B1

B-1. Evaluate $$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$ and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers. Here $[x]$ is defined to be the integer such that $[x] \leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.