In this problem you will use series to derive an important equation known as Eul

Question

In this problem you will use series to derive an important equation known as Euler's formula: e^ix = cos(x) + i sin(x)

a) expand the Maclauin series for the three functions e^x, cos(x), and sin(x); include at least the frist 6 terms in each of these series.

b) use substitution to obtain the Maclaurin series for the function e^ix; expand the series, including at least the first 6 terms.

c) recall that the imaginary number i has the property i^2 = -1. use this fact, together with parts a and b, to obtain the result e^ix = cos(x) + i sin(x).

Explanation / Answer

e^x=1+x+x^2/2!+x^3/3!+x^4/4!+x^5/5!+x^6/6!

cosx= 1-x^2/2!+x^4/4!-x^6/6!+x^8/8!-x^10/10!

sinx = x-x^3/3!+x^5/5!-x^7/7!+x^9/9!-x^11/11!

b)

e^ix=1+(ix)+(ix)^2/2!+(ix)^3/3!+(ix)^4/4!+(ix)^5/5!+(ix)^6/6!

=1+ix-x^2/2!-ix^3/3!+x^4/4!+ix^5/5!-x^6/6!

c)

e^ix=1-x^2/2!+x^4/4!-x^6/6!+i(x-x^3/3!+x^5/5!) (collecting odd and even terms separately)

=cosx+i sinx

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