Spoiler
Lets call the sum you want to calculate S, d-daily dosage,h-halflife, t-time(days).
S= d+d*0.5^(1/h)+d*0.5^(2/h)+....+d*0.5^(t/h)= d(1+0.5^(1/h)+0.5^(2/h)+...+0.5^(t/h))=[It's a geometric progression]=d(1-0.5^((t+1)/h))/(1-0.5^(1/h))
d=10
h=40
t=30
S=241.92 (w/o respect to any bioavailability)
That matches my numeric-only calculation in the prior comment with two caveats:
1) I applied the 1% bioavailability estimate to the 10mg dose to yield a 0.1mg bioavailable dihexa per daily dose.
2) The second significant digit in my calculation was 3 (2.3...), rather than your 4 (2.4...).
I'd still like to understand why the symbolic solution doesn't come out correctly for either integration or summation (from 1 so it shouldn't involve logs of numbers between 0 and 1 -- at least not in the variable being integrated or summed).
Edited by jabowery, 01 October 2014 - 11:04 PM.