Question

limlnn/n^a=0(a≧1)用极限定义证明

Answer 1

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.