Question

如何证明1+1/2^3 + 1/3^3 +1/4^3.+1/n^3小于5/4

Answer 1

1+1/2^3 + 1/3^3 +1/4^3……+1/n^3

≤1+1/8 + 1/27 +1/(4*3*4)……+1/[4*(n-1)*n]

=1+1/8 + 1/27+(1/4){ 1/(3*4)……+1/[ (n-1)*n]}

=1+1/8 + 1/27+(1/4)[ 1/3-1/4+……+1/(n-1)-1/n]

=1+1/8 + 1/27+(1/4)( 1/3-1/n)

=1+1/8+1/27+1/12 -1/4n

≤1+1/8+1/24+1/12 -1/4n

=5/4 -1/4n

≤5/4