Re: [考古] 95政大應數1

[LuisSantos (^______^)]

※ 引述《spss520 (剩下一個月啦)》之銘言：
: Let f[0,1]→R be continuous function and f(0)=0
:              1
: show  lim   ∫ f(x^n)dx=0
:       n→∞  0
: 謝謝大家


                 1
proof :   lim   ∫ f(x^n) dx
          n→∞  0

           1
       =  ∫  lim  f(x^n)  dx
           0  n→∞

           1
       =  ∫  f(lim  x^n)  dx    --------------(1)
           0    n→∞



       Since  f(lim  x^n)
                n→∞


            = f(x)  =  f(0)  =  0  ,  0 ≦ x < 1

                       f(1)        ,   x = 1


                         1
       ,  (1) =>   lim  ∫ f(x^n) dx
                   n→∞ 0

                    1
                =  ∫  f(lim   x^n) dx
                    0    n→∞

                    1
                =  ∫  f(0) dx
                    0

                    1
                =  ∫ 0 dx  =  0
                    0











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