Question

用柯西收敛准则证明这个数列收敛?要具体步骤,

Answer 1

对任意epsilon>0,存在正整数N = [1/epsilon]+1,使得对任意n>N,任意正整数p,有

|x(n+p)-x(n)| = 1/(n+1)!+1/(n+2)!+…+1/(n+p)!

< 1/(n+1)n+1/(n+2)(n+1)+…+1/(n+p)(n+p-1)

= 1/n-1/(n+p)