If we want to solve the equation $sec^2(x)$ for finding the all roots(real and complex), we have two ways: 1-Direct solving for $sec^2(x)=0$ 2-Or by convert the above equation to polynomial series as follow $a_{0}+a_{1}x+a_{2}x^2+a_{3}x^3......a_{n}x^n=0$
In the first way, there are no roots in real and complex plane, but in the second way there are infinity roots Which solution is the right one?
The Fundamental Theorem of Algebra can not be extended to power series.
Example:
$$e^z\neq 0$$ that is $$1+z+\frac{z^2}{2!}+\frac {z^3}{3!}+\cdots\neq 0$$