I need to prove that a non constant complex tends to infinity when the argument tends to infinity. I'd be happy for a hint.
thanks
I need to prove that a non constant complex tends to infinity when the argument tends to infinity. I'd be happy for a hint.
thanks
$\def\abs#1{\left|#1\right|}$*Hint*: You have by the triangle inequality $ \abs{\sum_{k=0}^n a_k z^k} \ge \abs{a_n}\abs{z}^n - \sum_{k=0}^{n-1}\abs{a_k}\abs z^k $
If the polynomial is $f(z)$, consider $g(w)=1/f(1/w)$ and prove that $\displaystyle\lim_{w\to 0} g(w)=0$. This means that $\displaystyle\lim_{z\to \infty} f(z)=\infty$.
This is a standard technique for handling infinity: bring it to zero using the inversion $w=1/z$.