My problem is Examine the series \sum_{n = 2}^\infty \frac{{n}^{lnn }}{{lnn}^{ n} } . I solved that:

Applying the Cauchy's test we have:

\sqrt[n]{\frac{{n}^{lnn }}{{lnn}^{ n} } } = \frac{{n}^{\frac{lnn}{ n} } }{lnn } and \lim_{n \to \infty} \frac{{n}^{\frac{lnn}{ n} } }{lnn } = 1 ( since let y = {n}^{\frac{lnn}{ n} we have lny = \frac{{lnn}^{2 } }{n }= \frac{2}{n }=0 while \lim_{n \to \infty}lnn= infinite => \lim_{n \to \infty}\frac{{n}^{lnn }}{{lnn}^{ n} }= 0 <1. Hence, the series converges. Is that right?