Question

求和：（3/1x2x4)+(5/2x3x5)+(7/3x4x6)+ +(2n+1)/n(n+1)(n+3)

Answer 1

∵(2n+1)/n(n+1)(n+3)=1/3(1/n-1/(n+1))+5/6(1/(n+1)-1/(n+3))
∴3/(1x2x4)=1/3(1-1/2)+5/6(1/2-1/4)
5/(2x3x5)=1/3(1/2-1/3)+5/6(1/3-1/5)
7/(3x4x6)=1/3(1/3-1/4)+5/6(1/4-1/6)
.
(2n+1)/n(n+1)(n+3)=1/3(1/n-1/(n+1))+5/6(1/(n+1)-1/(n+3))
故:3/(1x2x4)+5/(2x3x5)+7/(3x4x6)+.+(2n+1)/(n(n+1)(n+3))
=1/3(1-1/(n+1))+5/6(1/2+1/3-1/(n+2)-1/(n+3))
=n/(3(n+1))+5/6(5/6-1/(n+2)-1/(n+3))
=(2n/(n+1)-5/(n+2)-5/(n+3)+25/6)/6.