Chapter 7.3, Problem 102PE

In calculus, we can show that

$e^x=1+\frac{x}{1}+\frac{x^2}{2\cdot 1}+\frac{x^3}{3\cdot 2\cdot 1}+\frac{x^4}{4\cdot 3\cdot 2\cdot 1}+\mathrm{....}$ ,

$\cos x=1-\frac{x^2}{2\cdot 1}+\frac{x^4}{4\cdot 3\cdot 2\cdot 1}-\frac{x^6}{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}+\mathrm{......}$ , and

$\sin x=x-\frac{x^3}{3\cdot 2\cdot 1}+\frac{x^5}{5\cdot 4\cdot 3\cdot 2\cdot 1}-\frac{x^7}{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}+\mathrm{......}$

Write the first eight terms of the expansion of $e^{ix}$ to illustrate that $e^{ix}=\cos x+i\sin x$ .