用比值审敛法或根值审敛法判别下列级数的收敛性：(1)∑n=1∞3n/(n•2n)(2)∑n=1∞(n•3

用比值审敛法或根值审敛法判别下列级数的收敛性：
(1)∑_n=1^∞3ⁿ/(n•2ⁿ)
(2)∑_n=1^∞(n•3ⁿ)
(3)∑_n=1^∞[n/(2n+1)]ⁿ)
(4)∑_n=1^∞[n/(3n-1)]²ⁿ
(5)∑_n=1^∞1/[ln/(3+1)]ⁿ
(6)∑_n=1^∞3ⁿ•n/nⁿ
【正确答案】：(1)∵lim_n→∞(u_n+1)/u_n=lim_n→∞3ⁿ⁺¹/(n+1)2ⁿ⁺¹ •[(n•2)ⁿ/3ⁿ]=3/2﹥1 ∴∑_n=1^∞3ⁿ/(n•2ⁿ)发散 (2)∵lim_n→∞(u_n+1)/u_n=lim_n→∞(n+1)/3ⁿ⁺¹•(3ⁿ/n)=1/3﹤1 ∴∑_n=1^∞n/3ⁿ收敛 (3)∵lim_n→∞ⁿ√u_n=lim_n→∞n/(2n+1)=1/2﹤1 ∴∑_n=1^∞[n/(2n+1)]ⁿ收敛 (4)∵lim_n→∞ⁿ√u_n=lim_n→∞[n/(3n+1)]²=1/9﹤1 ∴∑_n=1^∞[n/(3n+1)]²ⁿ收敛 (5)∵lim_n→∞ⁿ√u_n=lim_n→∞1/ln(n+1)=0 ∴∑_n=1^∞1/[ln(n+1)]ⁿ收敛 (6)∵lim_n→∞[3ⁿ⁺¹•(n+1)!/(n+1)ⁿ⁺¹]•[nⁿ/3ⁿn!] =lim_n→∞3(n+1)/(n+1)•(1+1/n)ⁿ=3e﹥1 ∴∑_n=1^∞(3ⁿ•n)/nⁿ发散．