MathWorld gives the root linear coefficient theorem as
The sum of the reciprocals of roots of an equation equals the negative coefficient of the linear term in the Maclaurin series.
The theorem appears to me to be false as stated. For example, the equation $e^x = 0$ has no roots, yet (taking the Maclaurin series of $e^x$) the root linear coefficient theorem claims that the sum of the reciprocals of these nonexistent roots would be $-1$.
Is MathWorld missing some hypotheses? Or is there something happening in the complex plane that I'm not aware of? The MathWorld entry also says to see Vieta's formulas, but those are for polynomials and not Maclaurin series.
The only real information I could find from a Google search on "root linear coefficient theorem" was this statement (from Robert Israel of UBC):
This won't work in general for non-polynomials (e.g. try it for $p(x) exp(x)$. For a rational function such that $0$ is neither a root nor a pole, you want to take the sum of the reciprocals of the roots minus the sum of the reciprocals of the poles (again counting multiplicity).
O.K., so it won't work for non-polynomials, and for rational functions you have to include the poles.
But then why is MathWorld applying it to $\sin z/z$ in this "proof" that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$? (Start near eq. (18).)
The value $\zeta(2)$ can also be found simply using the root linear coefficient theorem. Consider the equation $\sin z=0$ and expand $\sin$ in a Maclaurin series $\sin z = z- \frac{z^3}{3!}+\frac{z^5}{5!}+ \ldots =0$ $0 = 1-\frac{z^2}{3!}+\frac{z^4}{5!}+ \ldots$ $ = 1-\frac{w}{3!}+\frac{w^2}{5!}+ \ldots,$
where $w=z^2$. But the zeros of $\sin z$ occur at $z=\pi, 2\pi, 3\pi, \ldots$, or $w=\pi^2, (2\pi)^2, \ldots$. Therefore, the sum of the [reciprocals of the] roots equals the [negative of the] coefficient of the leading term $\frac{1}{\pi^2}+\frac{1}{(2\pi)^2}+\frac{1}{(3\pi)^2}+ \ldots =\frac{1}{3!}=\frac{1}{6},$
which can be rearranged to yield $\zeta(2)=\frac{\pi^2}{6}.$
(This is where I ran across the root linear coefficient theorem in the first place.)
Could someone enlighten me with respect to these questions:
Is MathWorld just wrong?
Am I missing something here?
What is the correct statement of the root linear coefficient theorem?