f(x)=e^x-x求证(1/n)^n+(2/n)^n+...+(n/n)^n

Answer 1

令ak=[(n-k)/n]^n .k=n-1,...,3,2,1,0
--->ak = [1-k/n]^n = {(1+(-k)/n]^(n/-k)}^(-k)
--->n→∞时,lim(ak) = 1/e^k
--->n→∞时
lim[(1/n)^n+(2/n)^n+...+[(n-1)/n]^n+(n/n)^n]
=1+1/e+1/e²+...+1/e^k+...
=1/(1-1/e) = e/(e-1)
--->(1/n)^n+(2/n)^n+...+[(n-1)/n]^n+(n/n)^n＜e/(e-1)