1. You may have noticed the similarity of the expansion of the function e", sin(x) and cos(x). In fact, they are very closely related! Recall the imaginary value i = J1 . Substitute ix into the e\" function, simplifying the powers of i along the way. You get a very elegant, famous relationship between these three functions. Among other things, it gives meaning to raising a base to a complex exponent. Use this relationship to show i' = e'. As you study this, you'll come across deMoivre's theorem. How would you calculate a value for 3'\"? 2. One of the most famous series is the sum of the inverted squares: 2 1+ -1- + l + i + 215 + i+ i +... - \" . It may be surprising that the result is pi squared divided 4 9 16 36 49 6 by 6. The proof uses the Taylor polynomials for (sin(x))fx. (Hint: just take the Taylor formula for the sine function, then divide by x.) The real elegant steps involve a repeated product. For this you may have to do some web research: see Euler and/or s-series for some leads. The steps are surprisingly simple and elegant. You do not need to prove this, but explain how Euler did so. Euler was able to develop exact results for series of inverted integers raised to even powers, but not for odd powers. This is just one small part of the giant Riemann Hypothesis. What is it? Where does this series t in? Hint: This is called an s-series. Also, look up z-series. Again, the purpose here is for you to gain some insight into the hypothesis itself