Question

求下列级数的收敛半径和收敛域

Answer 1

(1) R = lim<n→∞>a<n>/a<n+1> = lim<n→∞> n!/(n+1)!
= lim<n→∞>1/(n+1) = 0;
(2) R = lim<n→∞>a<n>/a<n+1> = lim<n→∞> (n+1)2^(n+1)/(n2^n) = 2,
x = 2 时级数变为 ∑<n=1,∞>1/n 发散，x = -2 时级数变为 ∑<n=1,∞>(-1)^n/n 收敛，
收敛域 x∈ [-1, 1).
(3) 隔项级数，R^2 = lim<n→∞>a<n>/a<n+1> = lim<n→∞> (2n+1)/(2n-1) = 1,
收敛半径 R = 1.
x = 1 时级数变为 ∑<n=1,∞>1/(2n-1) > ∑<n=1,∞>1/(2n) = (1/2)∑<n=1,∞>1/n 发散，
x = -1 时级数变为 ∑<n=1,∞>-1/(2n-1) 发散，收敛域 x∈ (-1, 1).
(4) R = lim<n→∞>a<n>/a<n+1> = lim<n→∞>√(n+1)/√n = 1,
-1 < x - 5 < 1, 4 < x < 6,
x = 4 时级数变为 ∑<n=1,∞>(-1)^n/√n 收敛，
x = 6 时级数变为 ∑<n=1,∞>1/√n 发散，收敛域 x∈ [4, 6).