F=\dfrac{X^{n-1}}{X^n-2^n}

P=X^n-2^n

P=X^n-2^n =2^n \left( \left( \frac{X}{2} \right)^n-1 \right) =2^n\prod _{k=0}^{n-1} \left( \frac{X}{2}-e^{\frac{2ik\pi}{n}} \right) =\prod _{k=0}^{n-1} 2\left( \frac{X}{2}-e^{\frac{2ik\pi}{n}} \right)\\

\text{ on a : }P'=nX^{n-1} \text{ ainsi } F=\dfrac{1}{n}\dfrac{P'}{P}

P=\prod _{k=0}^{n-1} \left( X-2e^{\frac{2ik\pi}{n}} \right)

P=\prod _{k=0}^{n-1} \left( X-\alpha_k \right)^{m_k}

\alpha_k=2e^{\frac{2ik\pi}{n}}

m_k=1

\text{ on sait que (cours): } \dfrac{P'}{P}=\sum_{k=0}^{n-1} \dfrac{m_k}{X-\alpha_k}

\text{Conclusion: } F=\dfrac{1}{n}\sum_{k=0}^{n-1} \dfrac{1}{X-2e^{\frac{2ik\pi}{n}}}