Answer 2: Doodling

Assuming her arithmetic is accurate, she would never write the digit 8.

Reasoning

Now, this is equal to:
(1/10 + 1/100 + 1/1000 + 1/10000 + ...)
+ (1/100 + 1/1000 + 1/10000 + 1/100000...)
+ (1/1000 + 1/10000 + 1/100000 + 1/1000000...)
+ ...

Or in its decimal form
0.11111...+ 0.011111... + 0.0011111... + ...

Which is clearly
1/9 + 1/90 + 1/900 + 1/9000 + ...
= 1/9 (1 + 1/10 + 1/100 + 1/1000 + ...)
= 1/9 (1.11111...) = 1/9 x 10/9 = 10/81
= 0.123456790123456790123456790...

Hence, she will never write an 8.