證明級數收歛或發散...

If n>N=3^9>e^(e^2), then lnn>e^2, (lnn)^lnn > (e^2)^lnn=e^[ln(n^2)]= n^2
0<1/(lnn)^lnn < 1/n^2
so, Σ[n=2~∞]=Σ[n=2~N] 1/(lnn)^lnn +Σ[n=N+1~∞] 1/(lnn)^lnn
< Σ[n=2~N] 1/(lnn)^lnn+Σ[n=N+1~∞] 1/n^2
by comparison test, then Σ[n=2~∞] 1/(lnn)^lnn is convergent.