SOLUTION: Compute the sum and enter your answer as a common fraction: 1.11111111... + 0.11111111... + 0.01111111... + 0.00111111... + 0.00011111... + 0.00001111... + 0.0000

Question 1116977: Compute the sum and enter your answer as a common fraction:
1.11111111... + 0.11111111... + 0.01111111... + 0.00111111... + 0.00011111... + 0.00001111... + 0.00000111... + ....

Answer by ikleyn(52788) (Show Source): You can put this solution on YOUR website!
.

The number  1.111111 . . .   itself is the sum of the infinite geometric progression with the first term  a= 1 and the common ratio  0.1,  


so  1.111111 . . . =  = .



Next, the given sum is the sum of the geometric progression with the first term   and the common ratio of 0.1,


so this sum is equal to    =  = 1/0.81.


Answer.  The sum under the question is equal to  ,  or  .

Solved.

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