1.
如果An=n+(1/3)^n
Sn=n(n+1)/2+(1/3)×(1-1/3^n)/(1-1/3)
=n(n+1)/2+(1-1/3^n)/2
如果An=(n+1)/3^n
Sn=A1+A2+A3+……+An
=2/3+3/3^2+4/3^3+……+(n+1)/3^n
3Sn=2+3/3+4/3^2+……+(n+1)/3^(n-1)
兩式錯(cuò)位相減
2Sn=2+[(3/3-2/3)+(4/3^2-3/3^2)+……+(n+1)/3^(n-1)-n/3^(n-1)]-(n+1)/3^n
=2+(1/3+1/3^2+……1/3^(n-1))-(n+1)/3^n
=2+(1/3)×(1-1/3^(n-1))/(1-1/3)-(n+1)/3^n
=5/2-(n+5/2)/3^n
Sn=5/4-(n/2+5/4)/3^n
2.
An=1/(n^2+3n+2)
=1/[(n+1)(n+2)]
=[(n+2)-(n+1)]/[(n+1)(n+2)]
=1/(n+1)-1/(n+2)
Sn=A1+A2+A3+……+An
=(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+……+1/(n+1)-1/(n+2)
=1/2-1/(n+2)
=n/(2n+4)