<s>[INST] 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?
 [/INST] Formal power series can be added and then their area of convergence is limited by the most restricted one. 
So one easy way would be to 


*

*find separately expansions for $\{e^{x},\cos(x),\sin(x)\}$ 

*add (or subtract) them to each other

*put them under the same $\sum$ and then

*try to simplify the expression you get using algebra.
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