Convergence of $\sum_{n=3}^\infty \frac {1}{n \ln n}$ I know I have seen something similar and there is a telescoping trick to the convergence but it is eluding me.

For a decreasing function $a(x) \ge 0$, there is a theorem known as the Integral Test, which says that $\int_{m}^\infty a(x)$ converges if, and only if, $\sum_{n=m}^\infty a(n)$ converges. This is the (probably best) way of determining convergence or divergence here - you'll want to take $a(x) = \frac{1}{x\ln(x)}$.
