证明 1+1/8+1/27+....+1/n^3<29/24 要过程

因n^3-(n-1)n(n+1)=n>0
故n^3>(n-1)n(n+1)
故1/n^3<1/[(n-1)n(n+1)]=1/(2n)[1/(n-1)-1/(n+1)]
故1+1/8+1/27+....+1/n^3
<1+1/(1*2*3)+1/(2*3*4)+1/(3*4*5)+……+1/[(n-2)(n-1)n]+1/[(n-1)n(n+1)]-1/6+1/8
=1+1/2{(1/2)(1/1-1/3)+(1/3)(1/2-1/4)+……+[1/(n-1)][1/(n-2)-1/n]+[1/n][1/(n-1)-1/(n+1)]}-1/24
=1+1/2{1/2-1/[n(n+1)]}-1/24
=29/24-1/2[n(n+1)]
<29/24