fractions

Take the reciprocal of each:

0.11111111111 =1/9

0.01010101010 =1/99

0.000100010001 =1/9999

1/9 + 1/99 + 1/9999 =1213/9999

x = 0.11111111...
10x = 1.11111111...
10x-x = 1.1111111111...-0.111111111...
9x=1
x=1/9

y = 0.0101010101...
100y = 1.0101010101010...
100y-y = 1.01010101010...-0.01010101010101...
99y = 1
y = 1/99

z = 0.0001000100010001...
10000z = 1.000100010001...
10000z-z = 1.000100010001...-0.000100010001...
9999z = 1
z = 1/9999

$x+y+z\\ \frac{1}{9}+\frac{1}{99}+\frac{1}{9999}\\ \frac{1111}{9999}+\frac{101}{9999}+\frac{1}{9999}\\ \frac{1213}{9999}$