or where you looking for something more elegant in a form similar to ?
by the way is not the correct answer. I was only asking if you wanted something similar?
By the way do you mean
1.411411411 where 411 is always repeating or did you mean precisely 1.411411141144 as you wrote
if you meant 1.411411411 then the answer is
Yup, .411411 repeating forever is 411/999 which is 137/333. Notice that this works with every repeating decimal, just divide by a string of 9's of the same length as the repeating block. I found this out using a geometric series one day in proofs. Now it makes sense that any decimal with a repeating block is a rational number.
in addition try this more mechanical way to change repeating decimals into fractions:
Example: Transform .45123123123... into a fraction:
2. Multiply x by a power of 10, so that you get instantly repeating decimals:
3. Multiply 100x with a power of 10, so that the first block of the repeating digits is before the point:
4. Now substract:
Afterwards you may simplify this fraction – if it is possible!