Question

试证明(1/n)^n+(2/n)^n+(3/n)^n+......+((n-1)/n)^n<1/(e-1)

Answer 1

令An=(1/n)^n+(2/n)^n+(3/n)^n+......+((n-1)/n)^n 则
(n+1)^(n+1)a(n+1)=1^(n+1)+2^(n+1)+3^(n+1)+......+(n-1)^(n+1)+n^(n+1)
<n(1^n+2^n+3^n+......+(n-1)^n)+n^(n+1)=n^(n+1)(an+1) 所以
a(n+1)<n^(n+1)/(n+1)^(n+1)(an+1)=1/(1+1/n)^(n+1)(an+1)
注意到 (1+1/n)^(n+1)>e(因为递减趋于1） 所以
a(n+1)<1/e(an+1)=1/e+1/ean<1/e+1/e^2(a(n-1)+1)=1/e+1/e^2+1/e^2a(n-1)<
......<1/e+1/e^2+......1/e^(n-1)a2=1/e+1/e^2+......1/e^(n-1)*1/4
<1/e(1+1/e+1/e^2+......)=1/e*1/(1-1/e)=1/(e-1)