Analysis of $e^{ix}$ by Maclaurin Formula

AI Thread Summary
The discussion focuses on analyzing the expression $e^{ix}$ using the Maclaurin formula. It suggests substituting 'ix' into the known formula for $e^x$ to derive the result. Alternatively, it mentions using the power series for cosine and sine, multiplying the sine series by 'i', and then combining it with the cosine series. Both methods yield the same outcome, with the substitution method being quicker. The analysis ultimately leads to the conclusion that $e^{ix} = \cos x + i \sin x$.
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Analyze by Maclaurin formula:
$e^{ix}$
 
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Do you know the 'formula' for ex? If so, try substituting ix for x.
 
You could also do it by looking at the power series of cosx and sinx, and then multiply the sine power series by i and then add the power series of cos on to it, so you get cosx + isinx. Either way will get you the same answer, and the first method would probably be a little quicker.
 

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