I started with my brain and now I'm working backwards to try and figure out what I did....
This is not how you do science, starting with some conclusion you assume and then trying to justify it. It's how 'creation science' works and its how many hacks here work. You should start with the evidence and data and see where it takes you.
It should be a summation that included both the transcendental pi and the natural log hidden by the definition of a factored sum?
Odd form of summation?! $$ e^x= (\pi_{k=1}^{n}) K $$ (=) $$ e^x= (\sum_{k=1}^{n}) \frac{x^2}{n!} $$
This is just incoherent. Not only are you not doing science properly you cannot even do the maths properly. Why shoe horn in pi? Because it is transcendental?
Most numbers are transcendental, the algebraic numbers are staggeringly rare (they are as set of measure zero within the Reals). As it happens e is transcendental too, so if you want an transcendental number you already have one. In fact it is a proven theorem that if x is algebraic then $$e^{x}$$ is transcendental unless x=0.
Which is actually only a specific case of $$E^{2} = (mc^{2})^{2} + (pc)^{2}$$, the true mass-energy-momentum relationship in relativity.
There isn't a way for me to explain this but I can feel this next transformation is correct... In order to introduce a factorial we have to also introduce pi which could be where I messed up on the last one.
$${c}^{2}=\pi {\frac_{m!}^{E}}$$
There isn't a way to explain it because it is nonsense.
Last but not least to also accomplish this transformation to include both pi and factorial we have to bring with us our natural log given by the definition of factorial in summation form. So we can put that over our E.
$${c}^{2}=\pi {\frac_{m!}^{E^e}}$$
Summation form to reg is apparently not my strong suit...
Please tell me this is an attempt at a joke. Do you really think this is what physicists do, just randomly tape together different symbols because we feel like it?
Those are two different definitions of
Factorial, no where did I mention a definition of exponential, but I think the equation would represent that notion except where exponents occur.
Personally I prefer $$n! \equiv \Gamma(n+1)$$ where $$\Gamma(z) = \int_{0}^{\infty}e^{-t}t^{z-1}dt$$. And the definition of e comes from finding the y which satisfies $$\int_{1}^{y}\frac{dt}{t} = 1$$. From that you find a series expansion for $$e^{x}$$.
You solve two problems of equality by taking our imaginative ideas in science and smashing them with reality.
No, you solve science problems by examining reality and then drawing conclusions from that,
not making up your conclusions and then trying to justify them.