I have to prove that
is rational number.
I have one idea, I hope it's not crazy!
I have spoted that the sum of first three fractions are equal to 1:
I have then discovered that next five fractions are also equal to 1.
Continuing, we have that every odd number of fractions are equal to 1.
Considering that there are 99 fractions in that sum we get that because of whole sum is equal to 9:
I don't know how to prove that but it seems to me that there is a rule in that sum which I have described.