I am referring to the proof of fundamental theorem of algebra, i.e, nth order polynomial has n roots, starting with Liouville theorem in complex analysis, that says, if a function has no roots, it has to be bounded, and hence a constant, and if not a constant, then must have a root. Then divide by the (z-z(root)) and same argument follows upto which the function essentially becomes a constant…
What prevents us from extending this to an infinite series? Why cant we use the same argument to say that infinite series has infinite roots? That is not true in general, but why we cant extend this proof to that?
Best Answer
Because that argument uses not only the fact that the function $f$ has no roots but also the fact that$$\lim_{z\to\infty}\bigl\lvert f(z)\bigr\rvert=\infty,\tag1$$which holds for non-constant polynomial functions. And then $\frac1f$ is bounded and therefore then we can indeed apply Liouville's theorem. But $(1)$ doesn't hold in general.