Let \displaystyle f_{p}\left ( \beta \right )=\left ( \cos \frac{\beta }{p^{2}}+i\sin \frac{\beta }{p^{2}} \right )+\displaystyle \left ( \cos \frac{2\beta }{p^{2}}+i\sin \frac{2\beta }{p^{2}} \right )\cdots \\ \cdots\displaystyle\left ( \cos \frac{\beta(p-1) }{p^2}+i\sin \frac{\beta(p-1) }{p^2} \right )+\left ( \cos \frac{\beta }{p}+i\sin \frac{\beta }{p} \right )
then \displaystyle \lim_{n\rightarrow \infty } 1/f_{n}\left ( \pi \right )=.