How to write $\frac{1}{3\ln^23} + \frac{1}{4\ln^24} + \frac{1}{5\ln^25} +\dots$ as a short expression?

Question

I have a snippet of series that I have to note as a short expression:

$$\frac{1}{3\ln^23} + \frac{1}{4\ln^24} + \frac{1}{5\ln^25} +\dots$$

I am not sure if it would be more reasonable to write it as follows:

$\displaystyle\sum_{n=1}^{\infty} \frac{1}{(n+2)\ln^2(n+2)}$

or :

$\displaystyle\sum_{n=3}^{\infty} \frac{1}{n\ln^2(n)}\,.$

Would it even make a difference?