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→Arbitrary definition for using the imaginary square root of -1: upps
::<math>=1 + i \pi - \pi^2/2 - i \pi^3/6 + \pi^4/ 24 + \dots </math>
::<math>=1 + i \pi - \pi^2/2 - i \pi^3/6 + \pi^4/ 24 + \dots </math>
::<math>= \sum_{n=0}^{\infty} (-1)^n \frac{\pi^{2n}}{(2n)!} + i \sum_{n=0}^{\infty} (-1)^n \frac{\pi^{2n+1}}{(2n+1)!}</math>
::<math>= \sum_{n=0}^{\infty} (-1)^n \frac{\pi^{2n}}{(2n)!} + i \sum_{n=0}^{\infty} (-1)^n \frac{\pi^{2n+1}}{(2n+1)!}</math>
::<math>= \sin(\pi) + i \cos(\pi)</math>
::<math>= \cos(\pi) + i \sin(\pi)</math>
::<math>= -1 + i0 = -1</math>
::<math>= -1 + i0 = -1</math>
::Granted, it took two centuries to get all the details right (convergence, resorting of infinite sums, etc.), but that's the gist. Straightforward and beautiful. --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 16:04, 17 August 2015 (EDT)
::Granted, it took two centuries to get all the details right (convergence, resorting of infinite sums, etc.), but that's the gist. Straightforward and beautiful. --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 16:04, 17 August 2015 (EDT)