How to calc this sum

0.1 + 0.011 + 0.00111 + 0.0001111 + . . . =?
Sol
A=0.1 + 0.011 + 0.00111 + 0.0001111 + . . .
9A=0.9 + 0.099 + 0.00999 +0.0009999 + . . .
B= 0.1 + 0.001 + 0.00001 + 0.0000001 + . . .
9A+B=1+0.1+0.01+0.001+…=1.111….111=10/9
0.001/0.1=0.01
0.00001/0.001=0.01
0.0000001/0.00001=0.01
B=0.1/(1－0.01)=10/99
9A=10/9－10/99=(10*99－10*9)/891=900/891
A=100/891

0.1 + 0.011 + 0.00111 + 0.0001111 + . . . = ?

0.1
+ 0.011
+ 0.00111
+ 0.0001111
+ 0.000011111
+ 0.00000111111
+ 0.0000001111111
.
.
.
.................................
= 0.112233445566...

You are wrong, only the first four terms, the sum is

0.1 + 0.011 + 0.00111 + 0.0001111
= 0.1122211
> 1/9