<s>[INST] How to find the solutions of this equation? How to find all the solutions of the following equation? 
$1+\frac{x}{2!}+\frac{{{x}^{2}}}{4!}+\frac{{{x}^{3}}}{6!}+\frac{{{x}^{4}}}{8!}+...=0$ 
 [/INST] $$ \cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \dots $$
and
$$ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \dots $$
This means that for $x > 0$, $$\cosh \sqrt x = 1 + \frac{x}{2!} + \frac{x^2}{4!} + \dots $$
which clearly has no roots.  For $x < 0$, $$ \cos\sqrt{-x} = 1 + \frac{x}{2!} + \frac{x^2}{4!} + \dots $$
So you are after negative values of $x$ such that $\cos \sqrt{-x} = 0$
</s>