How to evaluate $\sum_{n=1}^{\infty} 180 \cdot \left(\frac25\right) ^n$ > A doctor has prescribed a medication for a patient. The patient will take 180ml of his medication every 6h and at the same time - every 6h his body will get rid of 60% of the total volume of medication within his body. What's the maximum volume of the medication that patient's body could receive? See the result below, but I am interested in figuring out how to get that by hand, even if you have a different approach. I am open for suggestions and any help is greatly appreciated. Thanks. $$S = \sum_{n=1}^{\infty} 180 \cdot \left(\frac25\right) ^n = 120$$

Simply use the results for the sum of an infinite geometric series! Recall that

$$1 + x + x^2 + ... = \sum_{k=0}^\infty x^k = \frac{1}{1-x} \;\;\; \text{provided} \; |x| < 1$$

Then notice: you can factor the $180$ out of the sum, as below:

$$S = \sum_{n=1}^\infty 180 \cdot (2/5)^n = 180 \sum_{n=1}^\infty (2/5)^n$$

Be careful now: the infinite geometric sum starts at the zeroth, not the first, term. But we can add $(2/5)^0 = 1$ to the summation to start at $n=0$ and then subtract it. Thus,

$$S = 180 \sum_{n=1}^\infty (2/5)^n = 180 \left(-1 + \sum_{n=0}^\infty (2/5)^n \right) = -180 + \sum_{n=0}^\infty (2/5)^n$$

Then, applying the formula above,

$$S = -180 + 180 \left( \frac{1}{1 - (2/5)} \right) = -180 + 180 \cdot \frac 5 3 = 120$$