Limits + Integration + Summation!

Calculus Level 5

We denote the two functions \( f(x) = \lfloor x \rfloor - x \) and \(\displaystyle g(x) = \lim_{n\to\infty} \frac{(f(x))^{4n}-1}{(f(x))^{4n}+1} \).

Find the value of \( \displaystyle \int \sum_{r=1}^{2014} (g(x))^r \, dx \).