Evaluate
There seemed to be some problem with stackexchange's math rendering but Ian corrected whatever error was there in the expression.Thanks
$$5050 \frac {\left( \sum _{r=0}^{100} \frac {{100\choose r}}{50r+1}\cdot (-1)^r\right) - 1}{\left( \sum _{r=0}^{101}\frac{{101\choose r}}{50r+1} \cdot (-1)^r\right) - 1}$$
The original definite integral that led to this was
$$5050\frac{\int_0^1(1-x^{50})^{100} dx}{\int_0^1(1-x^{50})^{101} dx} $$
I used binomail to arrive at the above expression.
I already know a technique by definite integration but I want one by sequence and series.
The answer is 5051