求(r+3)/[r(r+1)(r+2)]的前n项和Sn

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求(r+3)/[r(r+1)(r+2)]的前n项和Sn

求(r+3)/[r(r+1)(r+2)]的前n项和Sn
求(r+3)/[r(r+1)(r+2)]的前n项和Sn

求(r+3)/[r(r+1)(r+2)]的前n项和Sn
(r+3)/[r(r+1)(r+2)]=a/r+b/(r+1)+c/(r+2)
r+3=a(r^2+3r+2)+b(r^2+2r)+c(r^2+r)=r^2(a+b+c)+r(3a+2b+c)+2a
a+b+c=0
3a+2b+c=1
2a=3,a=1.5
b+c=-1.5
2b+c=-3.5
b=-2,c=0.5.5,a=1.5
Sn各项如上分解,并相互抵消后得:
Sn=1.5-2/2+1.5/2-2/(n+1)+0.5/(n+2)=1.25-2/(n+1)+0.5/(n+2)