Eulers identity history When Euler discovered/invented $e^{ix} = \cos(x)+i\sin(x)$. Did he doubt his calculations for a length of time? Was it Readily accepted by the mathematical community quickly or did they object at first? If so when was it finally accepted and why?

Work with power series was innovated by James Gregory and Isaac Newton almost a century earlier so the kind of arguments Euler presented in his 1748 _Introductio_ were easily digestible by his contemporary mathematicians. The _Introductio_ is full of calculations with power series a lot more complicated than the one you mentioned. I have seen no evidence that anybody doubted Euler's formula. His infinite product formula for the sine function was far more innovative; that also seems to have elicited no objections. For a detailed study of the formula see for example this acticle.