Recall the power series expansions from calculus. Derive Euler's formula e i = cos + i sin

Recall the power series expansions

from calculus. Derive Euler's formula e^iθ = cosθ + i sinθ formally from these three series expansions.

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e* = 1+x+ sin x = x cos x = 1 - - x² + + + 2! 3! 4! x x5 3! + 5! x² + 2! 4! J - 7! to 6! + + + + n! +(-1)^-1 x²n-1 (2n-1)! 2n +(-1)". + (2n)! + ..., and