Knowing the limit definition [math]e^k=lim_{n→∞}\left(1+\frac{k}{n}\right)^n[/math], explore what happens graphically when [i]k[/i]=[i]iπ[/i] for some finite values [i]n[/i].[br][br]Consider how when we multiply two complex numbers [math]a+bi=r_1\left(cos\alpha+isin\alpha\right)[/math] and [math]c+di=r_2\left(cos\beta+isin\beta\right)[/math], the product is [math]\left(a+bi\right)\left(c+di\right)=r_1\left(cos\alpha+isin\alpha\right)\cdot r_2\left(cos\beta+isin\beta\right)=r_1r_2\left(cos\left(\alpha+\beta\right)+isin\left(\alpha+\beta\right)\right)[/math].[br][br]Or by extension, [math]\left(r\left(cos\theta+isin\theta\right)\right)^n=r^n\left(cos\left(n\theta\right)+isin\left(n\theta\right)\right)[/math].[br][br]When [i]k[/i] is set to [i]iπ[/i] and [i]N[/i] is set to a "large" value, can you make sense of the resulting geometry? Does it help explain the value of [math]e^{i\pi}[/math]?[br][br]Drag the point representing k to other locations on the complex plane to explore other [math]e^k[/math] values.