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Taylor Series Expansion and derivation of the Euler Formula

Show that

(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.


Solution Preview

Following is the text part of the solution. Please see the attached file for complete solution. Equations, diagrams, graphs and special characters will not appear correctly here.

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) + ...

Solution Summary

I have provided a three-page solution to this problem. Using equation editor in MS word, every step of the solution is shown for the students to read and learn. This is an excellent practice question on the derivation of Euler formula and Taylor series in general.