General formula for $e^x+\cos(x)$, $e^x+\sin(x)$, $e^x-\sin(x)$, $e^x-\sin(x)$ I have been able to derive the formal series for these four functions:
$e^x+\sin(x) = 1+2x+\dfrac{x^2}{2!}+\dfrac{x^4}{4!}+\dfrac{2x^5}{5!}+\dfrac{x^6}{6!}+\dfrac{x^8}{8!}+\dfrac{2x^9}{9!}+...$
$e^x+\cos(x) = 2+\dfrac{x^3}{3!}+\dfrac{2x^4}{4!}+\dfrac{x^5}{5!}+\dfrac{x^7}{7!}+\dfrac{2x^8}{8!}+...$
$e^x-\sin(x) = 1+\dfrac{x^2}{2!}+\dfrac{2x^3}{3!}+\dfrac{x^4}{4!}+\dfrac{2x^7}{7!}+\dfrac{x^8}{8!}+...$
$e^x-\cos(x) = x+x^2+\dfrac{x^3}{3!}+\dfrac{x^5}{5!}+\dfrac{2x^6}{6!}+\dfrac{x^7}{7!}+\dfrac{x^9}{9!}+...$
Due to the missing term and the irregularity, I am unable to write the general formula for these series. I wish to find the general formula with compact sigma notation. Can you help with this?

If the terms in the series for the two functions you're adding or subtracting had the same numbering scheme, you could easily add or subtract them term by term within the same numbering scheme.
The reason you cannot do this with the usual series for these functions is that the $n$th term in the series for $e^x$ is the term in $x^n$ (numbering the terms starting with $n=0$), whereas for $\sin x$ the $n$th term is the term in $x^{2n+1}$:
$$ \sin (x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}  . $$
To fix this, we need a function $f$ over integers such that
$$
f(n) = \begin{cases}
\phantom{-}0 &&& n \equiv 0 \pmod 4,\\
\phantom{-}1 &&& n \equiv 1 \pmod 4,\\
\phantom{-}0 &&& n \equiv 2 \pmod 4,\\
-1 &&& n \equiv 3 \pmod 4.
\end{cases}
$$
This function can be written compactly using the formula $\Im(i^n),$ the imaginary part of the $n$th power of the imaginary unit.
Alternatively, $\sin\left(\frac\pi2 n\right)$ will do just as well.
You could even use an expression like
$\frac12(1 - (-1)^n)(-1)^{\lfloor n/2\rfloor},$
thereby avoiding all hints of imaginary numbers or trig functions.
But we could just define $f$ by cases as shown above in order to have less obscure jiggery-pokery in our formulas.
So we can write
\begin{align}
e^x &= \sum_{n=0}^\infty \frac{1}{n!} x^n, \\
\sin(x) &= \sum_{n=0}^\infty \frac{f(n)}{n!} x^n, \\
\cos(x) &= \sum_{n=0}^\infty \frac{f(n+1)}{n!} x^n. \\
\end{align}
Now it should not be hard to add or subtract these series term by term.
For example,
$$
e^x - \sin(x) = \sum_{n=0}^\infty \frac{1 - f(n)}{n!} x^n
$$
