Question

(2/3)+(5/9)+(8/27)+(11/81)+(14/243)+...+[(3n-1)/(3^n)]

求该式的值，谢谢~

Answer 1

设Sn=2/3+5/9+8/27+11/81+14/243+...+(3n-1)/(3^n)
∴1/3Sn=2/9+5/27+8/81+11/243+...+(3n-4)/(3^n)+(3n-1)/3^(n+1)
∴Sn-1/3Sn=2/3Sn
=2/3+3/9+3/27+3/81+....+3/3^n-(3n-1)/3^(n+1)
=2/3-(3n-1)/3^(n+1)+[1/3+1/9+1/27+...+1/3^(n-1)]
=2/3-(3n-1)/3^(n+1)+1/3[1-(1/3)^(n-1)]/(1-1/3)
=2/3-(3n-1)/3^(n+1)+1/2[1-1/3^(n-1)]
=7/6-(3n-1)/3^(n+1)-(9/2)/3^(n+1)
=7/6-(3n+7/2)/3^(n+1)
∴Sn=7/4-(6n+7)/(4×3^n)

Answer 2

　　解：原式=1-1/3+6/9-1/9+9/27-1/27+12/81-1/81+15/243-1/243+……+n/3^(n-1)-1/3^n
　　=1+6/9+9/27+12/81+15/243+……+n/3^(n-1)-【1/3+1/9+1/27+1/81+1/243+……+1/3^n】
　　=