TITLE: Intuitive Explanation why the Fundamental Theorem of Algebra fails for infinite sums
QUESTION [9 upvotes]: We know that for every polynomial of $n$.order Fundamental Theorem of Algebra guarantees $n$ complex roots.
Lets consider the complex exponential function $f(z)=\exp(z)$. As $f(z)$ is holomorphic, we are allowed to evaluate the power sum of $f(z)$.
$$f(z)=1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots$$ 
We also know that 
$$f(z)=\exp(z)=\exp(x+iy)=\exp(x)\exp(iy)=\exp(x)\cdot(\cos(y)+i\sin(y))$$
A zero of this expression would only be possible if $\exp(x)$ or $\cos(y)+i\sin(y)$ or both would vanish. But $\exp(x)>0$ and $|\cos(y)+i\sin(y)|=1$, hence $\exp(z)$ doesn't vanish. As the exponential is fully defined by the powersum with infinite radius of convergence, we can conclude that the powersum, despite of beeing of "polynomial like type" does not have any zeros.
I would be glad if someone could give me an intuitive explanation why the fundamental theorem of algebra is only true for finite polynomials.

REPLY [5 votes]: The infinity of roots is there (think of the roots of $\sin(z)$), but in the case of the exponential they escape to infinity.
Below a plot of the roots of the Taylor approximations of $e^x$ to degrees $5$, $7$ and $10$.

Also think of the approximations $\left(1+\dfrac zn\right)^n$, having a root of multiplicity $n$ at $z=-n$.