Question

一个交错级数,前2n项和与前2n+1项和均收敛,则此级数收敛?

Answer 1

a[k] ≥ 0,满足lim{n → ∞} ∑{1 ≤ k ≤ 2n} (-1)^k·a[k]与lim{n → ∞} ∑{1 ≤ k ≤ 2n+1} (-1)^k·a[k]均存在.
级数∑{1 ≤ k} (-1)^k·a[k]未必收敛.
例如a[k] = 1,有∑{1 ≤ k ≤ 2n} (-1)^k·a[k] = 0,∑{1 ≤ k ≤ 2n+1} (-1)^k·a[k] = 1.
二者都收敛,但级数通项不趋于0,级数不收敛.
若加上条件a[k] → 0,则可证明级数收敛.
因为lim{n → ∞} ∑{1 ≤ k ≤ 2n} (-1)^k·a[k] = lim{n → ∞} a[2n+1]+∑{1 ≤ k ≤ 2n+1} (-1)^k·a[k]
= lim{n → ∞} a[2n+1]+lim{n → ∞} ∑{1 ≤ k ≤ 2n+1} (-1)^k·a[k]
= lim{n → ∞} ∑{1 ≤ k ≤ 2n+1} (-1)^k·a[k],即两个极限相等.
此时易得lim{n → ∞} ∑{1 ≤ k ≤ n} (-1)^k·a[k]与二者相等.