[1+1/(1*3)]*[1+1/(3*5)]*[1+1/(4*6)]*...*[1+1/(11*13)]
有没有掉项?【 *[1+1/(2*4)] 】
如果原式=[1+1/(1*3)]*[1+1/(2*4)]*[1+1/(3*5)]*[1+1/(4*6)]*...*[1+1/(11*13)]
n*(n+2)+1=(n+1)^2
1+1/[n*(n+2)]=[n*(n+2)+1]/[n*(n+2)]=(n+1)^2/[n*(n+2)]
原式=[1+1/[n1*(n1+2)]*[1+1/[n2*(n2+2)]*[1+1/[n2*(n2+2)]*[1+1/[n3*(n3+2)]*...[1+1/[n11*(n11+2)]
=(1+1)^2*(2+1)^2*(3+1)^2*...*(11+1)^2/[(1*3)*(2*4)*(3*5)*...*(11*13)]
=2^2*3^2*4^2*...*12^2/(1*2*3^2*4^2*...*11^2*12*13)
=2^2*12^2/(2*12*13)
=2*12/13
=24/13
如果没掉项
n*(n+2)+1=(n+1)^2
1+1/[n*(n+2)]=[n*(n+2)+1]/[n*(n+2)]=(n+1)^2/[n*(n+2)]
原式=[1+1/(1*3)]*[1+1/(2*4)]*[1+1/(3*5)]*[1+1/(4*6)]*...*[1+1/(11*13)]/[1+1/(2*4)]
=(1+1)^2*(2+1)^2*(3+1)^2*...*(11+1)^2/[(1*3)*(2*4)*(3*5)*...*(11*13)]/(9/8)
=2^2*3^2*4^2*...*12^2/(1*2*3^2*4^2*...*11^2*12*13)*8/9
=2^2*12^2/(2*12*13)*8/9
=2*12/13*8/9
=64/39