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13

Part of 2010 Math Prize For Girls Problems

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Math Prize 2010 Problem 13

Source:

11/14/2010
For every positive integer nn, define SnS_n to be the sum Sn=∑k=12010(cos⁡k! π2010)n. S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \, \pi}{2010} \right)^n . As nn approaches infinity, what value does SnS_n approach?
trigonometry