I am now reading through M. Nair's classic paper on the lower bound of the least common multiple.
The first step in Theorem 1 is not obvious to me. I would greatly appreciate it if someone could help me understand why each part is true.
Here's the first step:
Consider, for $n\ge1$, the integral:
$$I = \int_0^1x^n(1-x)^n dx=\int_0^1\sum_{r=0}^{n}(-1)^r{n\choose r}x^{n+r}dx=\sum_{r=0}^n(-1)^r{n\choose r}\frac{1}{n+r+1}$$
I am confused how $r$ gets added; how the integral on the far left maps to the sum; and how the integral simplifies to the final result.
I suspect that this is straight forward calculus so I really appreciate any help. I suspect that this will teach me a very basic principle with integrals that will be incredibly helpful to me going forward. :-)