Quote:
Originally posted by JeffW:
Quote:
Originally posted by RocketX:
[b]The total is $1.002. I don't know why he would send .002 cents (charged in fractional cents?). Maybe having to send a check for such a dumb total inspired the clever check writing.
The series E(1/2^n) from 0 to infinity adds up to 2 with infinite terms.
e^(i*pi) is -1.
So it's 0.002 + -1 + 2 = 1.002
It seems to me his would be correct if the series started with n = 0. However, since the first term is n = 1, what you end up with is:

1/2 + 1/4 + 1/8 +. . . . + 1/inf = 1

e^xi = cos(x) + sin(x)*i = -1 + 0 * i = -1

So I end up with 2/10 of a cent.[/b]
Thats correct. The series in this case is converging to 1, not 2. Therefore its 0.002+e^ipi(=-1)+sum(1/2^n) = 0.002

I think rocketX made a mistake thinking that the series started at n=0, not n=1. if it were n=0, 1/(2^0) = 1. so it would be 1+0.5+0.25+.... = 2.