Determine whether the alternating series $\backslash (\backslash$sum$\_\{$n$=1\}^\{\backslash$infty$\}(-1)^\{$n$+1\}\backslash$frac$\{\backslash$ln n$\}\{$n$\}\backslash )$ converges or diverges.

Choose the correct answer below and, if necessary, fill in the answer box to complete your choice.

A$.$ The series does not satisfy the conditions of the Alternating Series Test but diverges by the Root Test because the limit used does not exist.

B$.$ The series does not satisfy the conditions of the Alternating Series Test but diverges because it is a p$-$series with $\backslash ($ p$=\backslash )$

C$.$ The series does not satisfy the conditions of the Alternating Series Test but converges because it is a p$-$series with $\backslash ($ p$=\backslash )$

D$.$ The series converges by the Alternating Series Test.

E$.$ The series does not satisfy the conditions of the Alternating Series Test but converges because it is a geometric series with $\backslash ($ r$=\backslash )$