(* CALCULUS 265 Honors Fall Semester 2000 8:00 to 8:50 MTTF Carver 0002 Instructor: Hentzel Office: 432 Carver Office Hours: 9:00-10:00 MTTF E-mail hentzel@iastate.edu phone 294-8141 Website http://www.math.iastate.edu/hentzel/honors.265 Thursday, November 2: Section 8.1 p555:24,31,45 p557:92,95,101 Main idea: Take as much time as you need to get as close as you want. Key Words: Series, Sequence, Converges harmonic series, power series, p-series, continued fraction, geometric series, limit, (in other words, all kinds of stuff) Goal: Learn a new technique that allows mathematics to think like we do. Who cares if we get it exact, we just keep going until we get it as close as we need it. Achilles can run 10 mph and a tortoise can go 1 mph. The tortoise has a 1 mile head start. Achilles cannot catch the tortoise because when he arrives at the place the tortoise was, the tortoise is beyond that. How long did it take for Achilles to catch the tortoise. How far did Achilles have to run? Find x. x = 1/2 + 1/4 + 1/8 + 1/16 + ... x = 1/3 + 1/9 + 1/27 + 1/81 + ... x = a + ar + ar^2 + ar^3 + ... x 1+Sqrt[1 + Sqrt[1 + Sqrt[1 + Sqrt[1 + ...... ]]]] . . . Sqrt[2] Sqrt[2] Sqrt[2] Sqrt[2] x = Sqrt[2] 1 x = 1 + ------------------ 1 1 + ---------------- 1 1 + --------------- 1 1 + -------------- 1 1 + -------------- 1 1 + -------------- 1 1 + -------------- etc ----------------- ----------------- 1 -----------------1/2 ----------------- 1/3 ----------------- 1/4 -----------------1/5 wt = n-1 Number of blocks above nth block -------------------- < 1-x >^<-x-> 1*(1-x) = (n-1)*x 1-x = n*x-x 1 = n*x x = 1/n So the overhang for the nth block is 1/n. x = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... How big does this get. This is the harmonic series.