國立政治大學95學年度應用數學系碩士班入學考微積分試題

Question

設f(x):[0,1]->R為一連續函數,試證:lim∫01f(xn)dx=0n->∞

不好意思,忘了一個條件:f(0)=0

Answer 1

By Mean Value Theorem for the Integral and f continuous on [0,k] 0<=k<1 we have
lim[∫0 to k] f(x^n)dx=lim f(cn^n)=f(lim cn^n)=f(0)=0 since0<=cn^n<=k^n->0 as n->∞
n->∞.......................... n->∞ ........ n->∞

we know that f is bounded on [0,1], and then the double limit
lim [∫0 to k] f(x^n) dx exist
n->∞ k->1
so that lim[∫0 to k] f(x^n) dx =lim lim [∫0 to k] f(x^n) dx
.......... n->∞......................... n->∞ k->1
=lim lim [∫0 to k] f(x^n) dx=lim 0=0
k->1 n->∞ .......................... k->1
Q.E.D.

Answer 2

因為剛考過

Answer 3

如何拿到95年各校碩士班的試題??