HM/MA 004, Assignment 9, due Friday, 8 November 2002, 10:00 AM

(Note on collaboration: If you do the homework in pairs or groups, please list the name(s) of your collaborator(s) on what you each turn in.)

1. (Considering sums and differences. (8 points) Use Leibniz's technique for finding the infinite sum of the reciprocals of the triangular numbers (Mathematical Expeditions, p. 131) to find the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... .

2. Sums and differences interpreted as calculus. (8 points) Explain and justify the remark in Mathematical Expeditions, p. 131, that "[t]his observation expresses, in a discrete, rather than continuous, way, the essence of the Fundamental Theorem of Calculus".

3. Doing Leibnizian integration. (8 points) Do Exercise 3.28 on p. 137 of Mathematical Expeditions.

4. The root concepts of Leibnizian integration. (8 points) Explain how Leibniz's "inverse tangent problem" approach to "quadrilateralization" (squaring curves) differently expresses "the essence of the Fundamental Theorem of Calculus".

5. The root concepts of Newtonian integration. (8 points) Explain how Newton's use of the "o-method" in Reader, section 12.A5 (p. 384) differently expresses "the essence of the Fundamental Theorem of Calculus".