Taylor Series Expansion and derivation of the Euler Formula
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(a) sin x = summation (0-infinity) (-1)^n x^(2n+1)/(2n+1)!
(b) cos x = summation (0-infinity) (-1)^n x^(2n)/(2n)!
Use the Taylor series expansion around the origin, f(x) = summation (0-infinity)[x^n/n!]f^n(0), and derive the power series expansions for sin x , cos x and e^x. Then write out the first few real and imaginary terms in the expansion for e^(ix), in order to demonstrate the well known complex exponential Euler formula, e^(ix) = cos x + i sin x by collection real and imaginary terms.
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eix = cos x + i sin x
Taylor expansion of eix around zero is
eix = where f (x) = eix
We will write out several terms in this expansion.
eix = 1 . f 0 (0) + (x/1) . f 1 (0) + ...
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