JanRabbit

the 1+q+q^2+q^3 part is (q^4-1)/(q-1). (it's a geometric series..) q-1=1-p-1= -p. so p(1+q+q^2+q^3) = p* (q^4-1)/-p = -q^4 + 1. which means (1-rate)^(1/4) = q. see? I don't think you need to take the derivative/use newton's method.

My dream from last night: went to take the SAT in this building in the middle of a forest area with tall redwoods, ferns, bird sounds, etc. i had a 5 course meal with me and i was supposed to eat the courses in between the SAT sections with the extra time I had. it was supposed to help me do well :P. The first course was this chicken leg (which tasted so nice; i don't eat chicken in real life) and lots of vegetables. there were shady-looking Hispanics next to me which made me feel a little hesitant, but i tried to enjoy my meal. Then the SAT started - the first section was math. I was in the middle of eating the chicken and my fingers were really greasy, so i had to fumble around a bit before commencing. many of the problems were confusing and suspiciously unofficial. i chalked that up to being due to the run-down building and delinquent looking test-takers. I went out of order, but managed to finish all the problems before time was called. but i wasn't confident in my performance so i committed myself to doing perfectly on the verbal and writing sections. my food bowl had wound up in the personal space of the kid next to me, and I could tell he was silently disturbed by this. from then on the dream kind of unraveled and i became more aware of the logical inconsistencies. yep, that's it .

Nice equation. I think you mean q is 1-p, not p-1 (it's positive right?). If it's 1-p then the p*(1+q+...+q^n) thing simplifies to -q^(n+1) +1. so q is actually just the (n+1)th root of (1-disease rate). for n=3 and disease rate = .1099, p=2.87%