limlnn/n^a=0(a≧1)用极限定义证明

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  • 证明 记 n^(1/n) = 1+h[n],有 h[n]>0,且

    n = (1+h[n])^n > C(n,2)(h[n])^2 = [n(n-1)/2](h[n])^2,

    于是,有

    0 < h[n] < √[2/(n-1)],

    于是,有

    lnn/n^a ≤ lnn/n = lnn^(1/n) = ln(1+h[n]) < h[n] < √[2/(n-1)].

    对任意ε>0,要使

    |lnn/n^a| ≤ lnn^(1/n) < √[2/(n-1)] < ε,

    只需 n > 2/(ε^2)+1,取 N=[2/(ε^2)]+2,则当 n>N 时,有

    |lnn/n^a| < √[2/(n-1)] < … < ε,

    得证

    lim(n→∞)lnn/n = 0.