limit{n->∞}(n^(n+1/n))/((n+1/n)^n)
=limit{n->∞}[n/(n+1/n)]^n*n*(1/n)
=limit{n->∞}[1/(1+1/n^2)]^n*limit{n->∞}n*(1/n)
=1/limit{n->∞}exp[n*ln(1+1/n^2)]*limit{n->∞}exp[(1/n)*lnn]
=1/limit{n->∞}exp(n*1/n^2)*limit{n->∞}exp(1/n)
=1/exp(0)*exp(0)
=1 不等于0
级数发散