设Sn=1+2+3+……+n(n∈N*),则f(n)=Sn/(n+7)Sn+1的最大值是多少

Sn=1+2+3+……+n=n(n+1)/2
f(n)=Sn/[(n+7)S(n+1)]
=n/(n+7)(n+2)
=1/(n+9+11/n)
∵n+11/n≥2√11,当且仅当n=√11时,取等号
又f(3)=3/50＜f(4)=2/33
所以f(n)在n=4处取得最大值f(4)=2/33

Sn=1+2+3+……+n=n(n+1)/2
S(n+1)=(n+1)(n+2)/2;
f(n)=sn/(n+32)s(n+1)=[n(n+1)/2]/[(n+32)*(n+1)(n+2)/2]
=n/(n+32)(n+2)=n/((n^2+34n+64)=1/(n+64/n+34)
由于
x+64/x>=2根号64=16 此时x=8
也就是n=8是n+64/n有最小值16
此时f(n)有最大值1/（16+34）=1/50