Hint needed(and exam tomorrow). Applications of calculus in calculating medicine dose

Hello everyone!

I am doing the following question:

The rate of decrease in the concentration of the drug in the bloodstream at time t hours is proportional to the amount x (mg) in the bloodstream at that time.

a) Show that $\displaystyle x=x_0e^{-kt}$, where k is a positive constant and $\displaystyle x_0$ is the size of the dose.

Ok, I did this part.

Then comes the difficult part.

b) Show that the amount of the drug in the bloodstream will never exceed $\displaystyle (\frac{x_0}{1-e^-{kT}})$

I did this

in the beginning of T=1

the amount of drug is

$\displaystyle x_0e^{-k}+x_0$

in the beginning of T=2

the amount of drug is

$\displaystyle x_0e^{-2k}+x_0e^{-k}+x_0$

Then I thought that this could be the sum of an infinite geometric progression. So the sum would approach the limit

$\displaystyle (\frac{x_0}{1-e^{-k}})$, but I'm missing the T!

Where am I going wrong?