Not everything can be verified by this trick, as we all know that the series is just approximation.
The infinite series is not an approximation and it is valid for all values of x, both real and complex. Don't factor out the x and set the initial condition that e^0 = 1 which gives c = 1 which is the first term x^0/0! and we the original series back -term by term. This is not a trick and integrating term by term in a convergent infinite series is valid process.
Again, e^x is actually equal to x^0/0! + x^1/1! + x^2/2! + ... to an infinite number of terms and is not an approximation.
@Ian Fowler but that will turn out to be a particular solution and not a general one.! To some extent that should be case for all trigonometric functions and exponent function. Then why we always include that (+C) term? The general solution for exponential should be like y=ce^x.