作业帮如何证明1+1/2^3 + 1/3^3 +1/4^3.+1/n^3小于5/4证明1+1/2^3 + 1/3^3 +1/4^3+1/n^3小于5/4
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如何证明1+1/2^3 + 1/3^3 +1/4^3.+1/n^3小于5/4证明1+1/2^3 + 1/3^3 +1/4^3+1/n^3小于5/4
如何证明1+1/2^3 + 1/3^3 +1/4^3.+1/n^3小于5/4
证明1+1/2^3 + 1/3^3 +1/4^3+1/n^3小于5/4
如何证明1+1/2^3 + 1/3^3 +1/4^3.+1/n^3小于5/4证明1+1/2^3 + 1/3^3 +1/4^3+1/n^3小于5/4
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
1+1/2^2+1/3^2+1/4^2+....1/n^2
<1+1/(1+2)+1/(2+3)+........1/(n-1+n)
=1+1-1/2+1/2-3.............+1/(n-1)+1/n
=2-1/n
=(2n-1)/n
当n>3时，1/n^3<1/2^(n+1)（可用导数证明）
（如果不用导数，我不知道怎么做）
所以
原式<1+1/8+1/16+1/32+……+1/2^(n+1)
<1+(1/8-1/2^n)/(1-1/2)
<1+(1/8)/(1/2)
=1+1/4
=5/4
1+1/2^2+1/3^2+1/4^2+....1/n^2
<1+1/(1+2)+1/(2+3)+........1/(n-1+n)
=1+1-1/2+1/2-3.............+1/(n-1)+1/n
=2-1/n
=(2n-1)/n
1+1/2^2+1/3^2+1/4^2+....1/n^2
<1+1/(1+2)+1/(2+3)+........1/(n-1+n)
=1+1-1/2+1/2-3.............+1/(n-1)+1/n
=2-1/n
=(2n-1)/n