Question Number 157593 by metamorfose last updated on 25/Oct/21
![let f:C→R z→min(y−[y],[y+1]−y) , y=Im(z) let w=e^(i((2π)/n)) , n∈N^∗ evaluate S_n =Σ_(0≤k<n) f(w^k )](https://www.tinkutara.com/question/Q157593.png)
$${let}\:{f}:{C}\rightarrow{R} \\ $$$$\:\:\:\:\:\:\:\:\:\:{z}\rightarrow{min}\left({y}−\left[{y}\right],\left[{y}+\mathrm{1}\right]−{y}\right)\:,\:{y}={Im}\left({z}\right) \\ $$$${let}\:{w}={e}^{{i}\frac{\mathrm{2}\pi}{{n}}} ,\:{n}\in{N}^{\ast} \\ $$$${evaluate}\:{S}_{{n}} =\underset{\mathrm{0}\leqslant{k}<{n}} {\sum}{f}\left({w}^{{k}} \right) \\ $$