Question

高数题求大佬用比值判别法或根值判别法判别下列级数的敛散性？

Answer 1

详细过程如图rt……希望能帮到你解决问题

Answer 2

(1) ∑<n=1,∞>1/(2n-1)! ,
ρ = lim<n→∞>a<n+1>/a<n>
= lim<n→∞>(2n-1)!/(2n+1)! = lim<n→∞>1/[2n(2n+1)] = 0,
则原级数收敛。
(2) ∑<n=1,∞>3^n/n^3 ,
ρ = lim<n→∞>a<n+1>/a<n>
= lim<n→∞>3^(n+1)n^3/[3^n (n+1)^3] = 3 < 1,
则原级数发散。
(7) ∑<n=1,∞>n^n/n! ,
ρ = lim<n→∞>[a<n>]^(1/n) >1, 级数发散。
或 ρ = lim<n→∞>a<n+1>/a<n>
= lim<n→∞>(n+1)^(n+1)n!/[n^n (n+1)!]
= lim<n→∞>(n+1)^(n+1)/[n^n (n+1)]
= lim<n→∞>(n+1)^n/(n^n) = lim<n→∞>(1+1/n)^n = e > 1,
级数发散。