Proof that sum is rational number

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.