8. (You will be required to Submit this problem.) Leonhard Euler used the Fundamental Theorem of Arithmetic1 to Show the surprising formula: 0" 1 1 EE=EI1ps for3>1, where Hp means the product over all prime numbers 1), so Thiss = (siss) (1 ss) (siss) (s is) 33' You don't need to prove that formula. (It took one of the greatest mathematical minds in history to discover it.) We can use this formula, however, to provide a different proof that there are innitely many prime numbers. (a) Fors>l,let 001 A = d. 03> l as I Using calculus, evaluate the improper integral to nd the value of A(s). (Use the limit denition of the improper integral.) (b) In the following picture, what are the areas of the four rectangular regions that intersect the curve y = 1/933? Are their areas combined greater or less than the area under the curve between a: = 1 and :r = 5? y If we continued the process of constructing such rectangles indefinitely1 what would be the area of the rectangle over the interval [n, 31+ 1]? How does A(s) relate to the innite series 200 i? Which one is bigger? Or are they the same size? 71:1 115 (c) Compute lim+ A(s) and explain why this tells you there are innitely many prime numbers. s>1