发道题,大家做下.

imath

证明: 1+1/2^3+1/3^3+1/4^3+...+1/n^3<5/4

疯不狂

n要什么条件？

imath

n∈N+

chsi

1/1*2*3+1/2*3*4+1/3*4*5+...+1/n*(n+1)*(n+2)
=1/2*(1/1*2-1/2*3)+(1/2)*(1/2*3-1/3*4)+……+(1/2)[1/n(n+1)-1/(n+1)(n+2)]
=1/2*[1/1*2-1/2*3+1/2*3-1/3*4+……+1/n(n+1)-1/(n+1)(n+2)]
=1/2[1/2-1/(n+1)(n+2)]

0<1/(n+1)(n+2)<1/2
所以0<1/2-1/(n+1)(n+2)<1/2
所以0<1/2[1/2-1/(n+1)(n+2)]<1/4
即1/1*2*3+1/2*3*4+1/3*4*5+...+1/n*(n+1)*(n+2)<1/4

即1+1/2^3+1/3^3+1/4^3+...+1/n^3<5/4

imath

用n^3<n*(n+1)*(n+2)放缩 ...

还有没有其它方法?

diameter

我还以为是e呢