We have the sequence, $a_{n}=\frac{1}{n \log ^{2} n}, n \geq 2$, where log is the logarithm to the base $2$ and let $A=\sum_{n=2}^{\infty} a_{n}$ be the sum of the sequence. We define a random variable and the corresponding distribution, $P(X=n)=p_{n}=\frac{a_{n}}{A}, n \geq 2$. Entropy or information of the random variable $X$ is defined as $H(X)=\sum_{n=2}^{\infty}-p_{n} \log p_{n},$ where $\log$ is the logarithm to the base $2.$ Which of the following is true about the entropy $H(X)?$
- $H(X) \leq 3$
- $H(X) \in(3,5]$
- $H(X) \in(5,10]$
- $H(X)>10$ but finite
- $H(X)$ is unbounded