Limit of $\lim _{ n->\infty }{ ({ z+{ z }^{ -1 }) }^{ n } } $ I'd like to compute the limit of $\lim _{ n->\infty  }{ ({ z+{ z }^{ -1 }) }^{ n } } $ for $z\neq 0$
My attempt was to use the Polar-Cor. representation for complexe numbers.So
${ (z+{ z }^{ -1 })^{n} }$=${ { (r }_{ 1 }(\cos { \theta  } +i\sin { \theta )+\frac { 1 }{ { r }_{ 2 }(\cos { \phi +i } \sin { \phi ) }  }  } ) }^{ n }$
and using Moivre's Formula we get
${ { r }_{ 1 } }^{ n }{ e }^{ i*n*\theta  }+\frac { 1 }{ { { { r }_{ 2 } }^{ n } }{ e }^{ i*n*\phi  } } $. How do I continue from here?

Let 
$$z+\frac1z=w.$$
Obviously, $w^n$ converges to zero if $|w|<1$ or to one if $w=1$.
By solving the above identity for $z$,
$$z=\frac{w\pm\sqrt{w^2-4}}2=\frac{re^{i\theta}\pm\sqrt{r^2e^{i2\theta}-4}}2$$ with $r<1$, or
$$z=\frac{1\pm i\sqrt3}2.$$
