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This is along the lines of Problem 9.8. in 'Concrete Mathematics' by Graham, Knuth and Patashnik.

Does any of the relation $\prec$, $\succ$ or $\sim$ exist between functions $f(n) =\displaystyle \sum_{k=0}^{n}k^{\lfloor \cos (k) \rfloor}$ and $g(n) =n^{\frac{3}{2}}$?

Both definitely diverge monotone to infinity, but I can't get my head around the rest.

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    It is a standard fact that $\pi$ is irrational. (I am assuming that as usual by $\cos(x)$ you mean the cosine function, where $x$ is measured in **radians**.)2011-05-31

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Since $\cos k \in (-1,1)$ for any positive integer $k$, $\left\lfloor {\cos k} \right\rfloor \in \{ 0, - 1\}$. Hence, $ f(n) = \sum\limits_{k = 0}^n {k^{\left\lfloor {\cos k} \right\rfloor } } \le \sum\limits_{k = 1}^n {k^0 } = n. $

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    @sigma.z.1980: Yes (more precisely, $f(n)=O(n)$ as $n \to \infty$).2011-05-31