高一數學題，請利用循環小數來解題謝謝 答案為 3967/8910?

A=0.474+0.04774+0.0047774+0.000477774+…
=(4/9)(0.999+0.09999+0.0099999+0.000999999+..........
+(3/9)(0.09+0.0099+0.000999+0.00009999+......
=(4/9)(1+0.1+0.01+0.001+......)-(4/9)(0.001+0.00001+0.0000001+0.000000001+......)
+(3/9)(0.1+0.01+0.001+0.0001+....)-(3/9)(0.01+0.0001+0.000001+0.00000001+....)
=(4/9)(10/9)-(4/9)(0.001)/(1-0.01)+(3/9)(1/9)-(3/9)(0.01)/(1-0.01)
=40/81-(4/9)(1/990)+(1/27)-(1/3)(1/99)
=(4400/8910)-(4/8910)+(330/8910)-(30/8910)
=4696/8910
=2348/4455

Sol
A=0.474+0.04774+0.0047774+0.000477774+…
B=0.101+0.01001+0.0010001+0.000100001+…
C=0.01+0.0011+0.000111+0.00001111+…
A=4B+7C
B=0.101+0.01001+0.0010001+0.000100001+…
D=0.1+0.01+0.001+0.0001+…=0.1/(1-0.1)=1/9
E=0.001+0.00001+0.0000001+0.000000001+…=0.001/(1-0.01)=0.1/99=1/990
B=D+E=1/9+1/990=111/990
9C=0.09+0.0099+0.000999+0.00009999+…
F=0.01+0.0001+0.000001+0.00000001+…=0.01/(1-0.01)=1/99
9C+F=0.1+0.01+0.001+0.0001+…=0.1/(1-0.1(=1/9
9C=1/9-1/99=10/99
C=10/891
A=4B+7C=444/990+70/891=2348/4455