原式=(5/9)*[1*(4/9)^0+2*(4/9)^1+3*(4/9)^2+......................]
[]內拆開
(4/9)^0+(4/9)^1+(4/9)^2+(4/9)^3+(4/9)^4+..............
(4/9)^1+(4/9)^2+(4/9)^3+(4/9)^4+..............
(4/9)^2+(4/9)^3+(4/9)^4+..............
(4/9)^3+(4/9)^4+..............
.....
....
...
..
.
找到規則了嗎？
[]內等於1/(1-4/9)+(4/9)/(1-4/9)+(4/9)^2/(1-4/9)+...............
分母與[]外的(5/9)消掉
剩下[]裡面1+(4/9)+(4/9)^2+(4/9)^3+.........
最後..原式=1/(1-4/9)=9/5

應該沒算錯....用文字解題真不習慣