Question

高等数学 用比值审敛法判定下列级数的敛散性 求指教

Answer 1

(5)
令Un=2^n/n!
Un+1=2^(n+1)/(n+1)!
lim n→∞ [2^(n+1)/(n+1)!] / [2^n/n!]
=lim [2^(n+1)n!] / [2^n (n+1)!]
=lim 2/(n+1)
=0
所以该级数收敛。
(6)
令Un=(3n-1)/3^n
Un+1=(3n+2)/3^(n+1)
lim n→∞ [(3n+2)/3^(n+1)] / [(3n-1)/3^n]
=lim [(3n+2) 3^n] / [(3n-1) 3^(n+1)]
=lim (3n+2)/[3(3n-1)]
分子分母同除n
=lim (3+2/n) / [3(3-3/n)]
=3/9
=1/3＜1
所以该级数收敛。