# I am floored again

Let $$m$$ and $$n$$ be positive integers such that $$\textrm{gcd}(m,n)=1$$. Further suppose that $$m$$ is even and $$n$$ is odd. Then what is the value of: $\frac{1}{2n} + \sum_{k=1}^{n-1} \left [ (-1)^{ \lfloor\frac{mk}{n}\rfloor} \left\{ \frac{mk}{n}\right\} \right]$

Details and Assumptions:

• $$\left\{\frac{mk}{n}\right\}$$ denotes the fractional part of $$\frac{mk}{n}$$.

• $$\lfloor x \rfloor$$ denotes the greatest integer $$\leq x$$.

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