Difficult math Q!?

Question

http://www.chizeng.com/misc/math/awesomeqs/8.png

Please also show your work. You may refer to the Fibonacci sequence, the geometric sequence sum formula, or any other means of obtaining the answer. This problem is from a sample GPML test.

Answer 1

It's b) 10/89. To see this, this series can be taken apart and put back together again in the following way:
First subtract the series 1/10 + 1/10^2 + 1/10^3 + 1/10^4 + ....
Then subtract the series (1/10)(1/10)(1/10 + 1/10^2 + 1/10^3 + ....)
Then subtract the series (1/10)(1/10^2)(1/10 + 1/10^2 + 1/10^3 ..)
Then subtract the series (2/10)(1/10^3)(1/10 + 1/10^2 + 1/10^3 ..)
Then subtract the series (3/10)(1/10^4)(1/10 + 1/10^2 + 1/10^3 ..)
Then subtract the series (5/10)(1/10^4)(1/10 + 1/10^2 + 1/10^3 ..)
A little inspection shows that if S is the sum of the original series, then the following must be true:
(1/9) + (1/10)(1/9)S = S
From this we see that S = 10/89

Answer 2

I believe the answer is B. 10/89